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QUESTIONS

MATHEMATICS

JOHN C. SMITH,

AUTHOR OF " THE CULMINATION OF THE SCIENCE OF LOGIC.

PUBLISHED BY

HT5RF5ERT C. SMITH,

16 COURT ST., BROOKLYN, N. Y.

SEEN BY
PRESERVATION

SERVICES

DATE

Electrotyped by R. HARMKE SMITH & SONS. 82 Beekman St., New York.

PREFACE.

THIS book is an outcome of one recently published,
entitled "The Culmination of the Science of Logic." .

The striking analogy between the necessary forms of
the process of reasoning and the simplest forms of geom
etry, exhibited by the author in that book, led him to
the reflection that perhaps the processes of geometry
would .have been greatly simplified, and its operations
therefore more easily performed, if the regular triangle
and tetrahedron, instead of the square and cube, had
been adopted as units of measure of surface and solidity.

The determination of this question, as the author was
well aware, required a better acquaintance with such
operations and processes than he possessed, he being but
a tyro, in the secondary sense of that word, in. mathe
matical science. A tyro may, however, ask questions ;
but to ask a question without at the same 'tittle 'giving, «
some reasons for the asking, would be to obtain Jf&6 it
but slight and insufficient attention.

The author thereupon commenced an investigation ,p£v
the subject for the purpose of finding such reasons, if
any there were, other than those which had led him up
to the question, intending to submit the question, with

IV PREFACE.

the reasons, to those qualified to consider and determine
it, through the columns of some scientific journal. But
the field of investigation widened as he advanced, ^and
further questions suggested themselves, until finally the
results, after many prunings down, took shape as set
forth in the following pages. Some of such results may,
perhaps, have no relevancy to the main question or any
other of the propounded questions, but they are given
because they may possibly be of service in further inves
tigation of the subject, if it be deemed worthy of pursuit.

The main question, as before indicated, in so far, at
least, as relates to the substitution of the triangle for the
square, must have suggested itself to the mind of almost
every thoughtful student of the science. The author
himself (who, however, can hardly be said ever to have
been a student of the science) observed and considered
it in years long gone by, but with reference to the tri
angle only. Such consideration was necessarily very far
from thorough, and resulted in his inability to see that
any advantage would be gained by the substitution.
Perhaps the consideration by others may have been in
like manner limited and, therefore, insufficient and at
tended with the like result.

It may be, however, that the question has been thor
oughly considered with reference to both the triangle and
the tetrahedron, and if so, then the conclusion arrived at
must have been adverse, or that no advantage would
result from the substitution. If such be the case, the
fact has never come to the knowledge of the author. The
subject matter and the results of his investigation are all
new to him.

PREFACE. V

But in either case he feels confident that the subject
has never been considered in the light of the analogy
referred to, and such analogy, if in form only, seerns to
him to give sufficient importance to the question to call
for an attentive and exhaustive consideration in the first
case, or a reconsideration in the second.

The author puts forth the book with great diffidence,
but is impelled by a sense of duty. It will be -manifest
from the tone of his questions what his opinion in respect
to each is, but such opinions — except in so far as they
are supported by processes shown, whereby the opera
tions of evolution and involution to the third degree
may be readily performed — are founded only upon in
tuition (in the literal sense of that word), and have not
the strength of convictions. If they shall be confirmed
by competent authority, then a benefit will have been
conferred upon mankind, which, but for the publication
of the book, would, perhaps, have forever remained
unknown.

If, however, on the other hand, they shall not be con
firmed, then it would seem that nature, while conforming
her processes in the two sciences to each other on the
faces of her simplest regular figures in a most wonderful
order and symmetry, has not designed the lines of such
figures as her chosen paths of investigation and reason
ing in mathematical science, but has rather made choice
of devious paths along the lines of complex, although
still regular, figures by which such investigation and
reasoning can be more advantageously pursued, thus
making order and symmetry merely formal, and of no
significance, value or effect.

VI - PREFACE. ,«

The treatise is divided into two parts, the main part
and an appendix. The appendix pertains, perhaps, more
to the science of Logic than to that of Mathematics. It
will be found to be illustrative and fully corroborative of
the doctrine of sorites as unfolded in the author's first
book, uThe Culmination of the Science of Logic," and
in this aspect is properly an appendix. But it is also
illustrative of the relations of the parts of certain geo
metrical figures described in the main part of the trea
tise, in respect as well to their construction, or rather
the combinations' of their parts, as to their analogy to
compound processes of reasoning, and in this respect it
is supplementary and entitled to be regarded as a part
of the treatise.

The author can hardly indulge the feeling of assur
ance that he has made no mistakes, but he is confident
that, if any are found, they will not be serious nor such
as to affect unfavorably the general design of the work.
He makes no apology for the diffuseness of his style, or
the profuseness of his illustrations. He has confessed
himself to be a tyro, and from such could not be ex
pected the conciseness and precision of an expert.

If the presentation of the subject be such as to engage
attention and draw forth answers to the questions, then,
whatever such answers may be, the object of the author
will have been accomplished.

BROOKLYN, December 4, 18S9,

CORRIGENDA.

Page 28, 3d line. Insert "produced" after "altitude".

31, 15th line. Insert "but produced," after "triangles"

and "produced" after "altitude".
35, 5th and 7th lines. Insert " combined" before " sorites"

in each line.
41, 2d line from bottom. Insert " each " after " will "

and strike out "adjacent".
72, 5th line. After " her," insert " when the number of

such edge or root does not exceed 10005,".
94, 3d line. After "number," insert "when the number

of such root does not exceed 10005,".
130. Strike out 2d and 3d flfl.

132. Strike out 2d fl. llth line from bottom. Insert "but"

before " with ".

133. 9th line. After "found" insert "by investigation (as

well £s by reasoning)".

134. On folded sheet following, mark symbols " N " desig

nating two points in 3d horizontal line of Fig. 3b
and " X'," two points in like line from bottom of
Fig. 196 " to he considered as in full-faced type ".

130, 10th line from bottom. Substitute " original " for
" principal ".

140, 10th line. For " octrahedron " read " octahedron ".
last line. Substitute " original " for "principal".

153, 4th line. Strike out "each of".

154, 4th line from bottom. Insert "exterior" before

SEE POSTSCRIPT.

QUESTIONS

MATHEMATICS

§ 1. The square and the cube have served from time
immemorial as units of measure of surface and solidity.
The square was undoubtedly adopted, when the occasion
for its use first arose (probably for the measurement of
land, and hence the name geometry), because of the uni
formity of its parts and its apparent simplicity ; and the
cube, when the occasion for its use first arose, was natu
rally adopted for the same qualities and as being modeled
upon the square. That they are the best adapted of all
geometrical figures to serve as such units of measure for
all the ordinary practical purposes of every -day life, there
can be no question. They have the appearance, at first
sight, of being simple figures, and are very readily com
prehended. But they are, in fact, both complex figures,
and it would seem, that when the study of geometry
came on to be pursued either for its own sake, inde
pendently of its practical application, or for that of
the higher purposes to which it is applicable, and espe
cially when questions involving the third dimension of
space came under consideration, the fact that the units
of measure theretofore in use for ordinary purposes only,

8 QUESTIONS IN MATHEMATICS.

were not the simple forms suggested by nature, would
have been recognized, and the question considered,
whether they should not be discarded for such higher
purposes, and the two and only simple and regular forms
of plane surfaces bounded by straight lines, and solids
with plane surfaces (the regular triangle and regular
tetrahedron), adopted in their stead, not only as actually
in accord with nature, but also as being more likely to
lead to simplicity in the operations and processes to be
founded upon them, than the square and cube in the
operations and processes founded upon them.

Whether they would or not lead to such greater sim
plicity is the main question, which it is the object of this
book to submit to those qualified to consider and deter
mine it.

That two units of measure of space may be concur
rently in use, the one apparently simple but in fact com
plex and em'ployed for the ordinary purposes of life,
and the other the true, simple unit in actual accord with
nature and employed for higher purposes, is evidenced
by the use of two different units of measure of time,
viz. : the solar day corresponding to the former, and the
sidereal day corresponding to the latter.

§ 2. The square in its simplest analysis is composed
of two equal right-angled isosceles triangles. It is, in
fact, if the expression may be allowed, a double unit, con
sisting of two right-angled triangular units combined. It
alone, or in conjunction with the linear unit upon which
it is described, can be applied only to the measurement
of areas in the form of right-angled parallelograms, all

QUESTIONS IX MATHEMATICS.

sides of which are accessible throughout their whole
extents. Areas, the angles of which are not all right
angles, and right-angled areas the boundaries of which
are not accessible throughout their whole extents, can
only be measured by means of the triangular unit. But
the triangular unit, which is the half of a square, is
irregular, consisting of three lines, of which two only are
equal to each other, the third being incommensurable,
except in power, with each of the other two, and three
angles, of which also two only are equal to each other,
the third being double each of the other two.

On the other hand, the regular triangle has all its
lines and angles equal to each other respectively, being
both equilateral and equiangular.

The. hypothenuse of the right-angled isosceles triangle
is, as will be hereinafter shown, the real linear unit
upon which the square is constructed. By means of it
as the radius of a circle and invariably representing
unity, and the varying dimensions of the other two
sides of all possible right-angled triangles formed upon
it within the circle, are all angles measured, but only
by means of squares considered as formed upon the sides
of such triangles. All angles may be measured directly
by the regular triangle and without recourse to squares,
as will be hereinafter shown.

The diameters of the inscribed and circumscribed cir
cles of the square are incommensurable with each other,
except in power, and one only, that of the inscribed
circle, is commensurable with, or rather equal to, the alti
tude of the square ; but the diameters of the inscribed
and circumscribed circles of a regular triangle are not

10 QUESTIONS IN MATHEMATICS.

only commensurable with each other, but both are also
commensurable with, though neither equal to, the alti
tude of the triangle.

A square can only be increased in area and its form
preserved by extending equally two of its sides having
a common point, each in its direction from such point,
and drawing from the extremities of such produced
sides two lines parallel to its other two sides until the
lines so drawn meet at a common point ; but a regular
triangle may be increased and its form preserved by a
like extension of any two of its sides, and connecting
the extremities of the produced sides by a single straight
line which will be parallel to its third side.

AVhile it is not of the very essence of a unit of meas
ure that it should have all its similar parts equal to each
other, it is, nevertheless, of the highest importance as
conducive to simplicity in application and calculation
that such should be the case, and the square was adopted
because of its conformity to this seeming requirement.
But the square is not the real unit.

Should not, therefore, the triangle, which is the real
unit, be uniform in respect to all its similar parts, and
instead of the irregular right-angled isosceles triangle
be the regular equilateral and equiangular triangle ?

§ 3. The cube is a highly complex figure, being com
posed in its simplest analysis of five figures, of which
one only is regular, viz. : a regular tetrahedron, the
nucleus of the cube and wholly hidden therein, and
having for its six edges diagonals of the six faces of the
cube. The remaining four figures are irregular, viz.:

QUESTIONS IN MATHEMATICS. 11

equal right-angled tetrahedra, each having as its base a
regular triangle equal to each of the faces of the hidden
regular tetrahedron, and each superposed on one of such
faces. The volume of the cube is three times that of the
included regular tetrahedron, each right-angled tetrahe
dron being one-half the volume of such regular tetra
hedron and one-sixth the volume of the cube.

The diameters of the inscribed and circumscribed
spheres of the cube and regular tetrahedron are incom
mensurable or commensurable with each other and with
the altitude of the cube and tetrahedron respectively, in
like manner as those of the inscribed and circumscribed
circles of the square and regular triangle, relatively to
each other, and to the altitude of the square and triangle
respectively.

A cube can only be increased in volume and its form
preserved by superposing upon each of three of its faces
having a common point a parallelopipedon of equal face,
and then filling up three parallelepipeds of equal length
and thickness, and after them a remaining cube, but a
regular tetrahedron may be increased and its form pre
served by adding uniformly to any one of its faces.

§ 4. A regular tetrahedron may, by four sections
beginning in the middle of four of its edges, and made
parallel to the opposite faces respectively, be divided
into five figures, all of which will be regular, viz. : four
equal regular tetrahedra, and the fifth and interior figure
a regular octahedron. If the original figure be considered
as of the edge of 2, the five figures into which it is divided
will each be of the edge of 1. The octahedron is equal

12 QUESTIONS IN MATHEMATICS.

in volume to the four tetrahedra combined. The original
figure is therefore equal in volume to eight tetrahedra of
the edge of 1.

If now the original figure be considered not as actually
divided, but as marked on its faces with the lines of the
division, there will be four faces of the octahedron visible
and four invisible. If, upon the four visible faces of the
octahedron there be superposed four regular tetrahedra
of the edge of 1, the resulting figure will be in the form
of an eight-pointed star (which may be called an oct'as-
tron), consisting of two equal inter volved regular tetra
hedra (edge 2), to "both of which the interior and now
wholly hidden octahedron is common. The points are
the extremities of the axes of the oct'astron, of which
there are four. The oct'astron is equal in volume to
twelve tetrahedra (edge 1), viz. : the octahedron equal to
four, and the eight superposed upon its faces. If six of
these tetrahedra, three of the original figure and three of
those superposed as above described, be cut off, leaving
two, the points of which are the opposite extremities of
any one of the axes of the oct'astron, there will remain a
figure consisting of the octahedron with two tetrahedra
attached to opposite faces thereof, and equal in volume
to the six detached tetrahedra. If now there be three
planes passed through the octahedron in line with its
edges extending from one tetrahedron to the other, and
through the centre of the octahedron (but in such man
ner that the edges between the visible faces of the octa
hedron and the two tetrahedra be considered as not sev
ered), the figure will be divided into two equal parts,

QUESTIONS IN MATHEMATICS. 13

each consisting of a regular tetrahedron with four irregu
lar equal right-angled tetrahedra attached, one by one of
its faces and the other three each by one of its edges.
If the two parts be now considered as put together again
and the axis reunited at the centre, so that it will hold
the two parts relatively in the same position (the figure
being considered as standing or held so that the axis
shall be vertical), the six right-angled tetrahedra, at
tached by their edges, may be folded over (three upward
and three downward) on the three visible faces of each
of the two regular tetrahedra, and the resulting figure
will consist of two perfect cubes connected together at
the middle point of an axis running through. a diagonal
of each. The six detached tetrahedra being equal in
volume to the octahedron and two tetrahedra so changed
in form, may be considered as changed also in form to
two other similar cubes, although they cannot be as sim
ply dissected, and the parts put together in the new
form. Thus the oct'astron is equal in volume to four
such cubes, and each such cube is equal in volume to
three tetrahedra (edge 1). Each cube has the diagonal
of each of its faces equal to the edge (1) of its included
tetrahedron, and each edge of the cube is therefore equal
to VJ) = .7071.

Does it not clearly appear from the foregoing analysis
of the regular tetrahedron (the simplest form of three
dimensions in nature), the building up or completion
thereon, or rather upon the octahedron therein contained,
of the oct'astron and the subsequent dissection of the
latter for the purpose of finding its contents in the com-

14 QUESTIONS IN MATHEMATICS.

plex form of cubes, that the diagonal is the real linear
unit upon which the squares, the bounding faces of the
cubes, are constructed ?

§ 5. The triangle of geometry is the analogue of the
syllogism of logic in respect to form, and the quadrilateral
is the analogue of the simple sorites (syllogism of four
terms) in either the ascending or descending direction of
the process of reasoning, but limited in each case in so
far as the notion of space comes under consideration to
space of two dimensions. The tetrahedron is the ana
logue of the sorites in both the ascending and descending
directions of the process of reasoning combined, ascend
ing first from subject to predicate on two faces of the
tetrahedron, and then descending from predicate (now
become subject) to subject (now become predicate) on the
other two faces, or vice versa. Such combined sorites
may, however, be considered as in either direction
throughout, the first progressive and the second regress
ive, or vice versa.

In the following figures, each equal to the other and
each in the form of a quadrilateral composed of two
regular triangles, let the full continuous lines repre
sent the only lines which can be actually measured, and
the dotted, and partly dotted lines, those which are the
results of processes of reasoning, and let the arrows
introduced in dotted lines indicate the directions in which
the points toward which they are directed can be seen
from the points from which they are directed respect
ively, and let the ultimate points, N in the first figure
and X in the second, be regarded as inaccessible.

QUESTIONS IN MATHEMATICS.

Fig. I.

Fig. 2.

The letters by which the points of the figures are
designated are the symbols adopted by the author to
represent the terms of the sorites, their logical significa
tions (as such terms were named also by the author)
being as follows (reading the first column of symbols
downward and the second upward in each case in con
nection with the logical significations) :

Descending

LOGICAL SIGNIFICATIONS.

Ascending

from

Subject

Maximus term

Predicate

Major-middle term

/ rum

Min or- middle term

Predicate

Magnus term

Subject

Each triangle is the analogue of a syllogism ; each
quadrilateral the analogue of a simple sorites, viz. : that
composed of triangles 1 and 2 taken together, without the
diagonal, in the descending direction of the process of
reasoning, and that composed of triangles 3 and 4 taken
together, without the diagonal, in the ascending direction.
The diagonal in each case represents the unexpressed
conclusion of the first, which is the unexpressed premise
of the second of a series of two syllogisms into which
each sorites may be fully expanded.

QUESTIONS IIST MATHEMATICS.

The sorites are as follows, those in the descending
direction being as shown in Fig. 1, and those in the
ascending direction as in Fig. 2.

Progressive descending.
X comprehends J,
J comprehends D,
L) comprehends .N J

'. X comprehends !N .

Progressive ascending.
K" is comprehended in D,
D is comprehended in J,

J is comprehended in X j
'. N is comprehended in X.

Regressive ascending.

J is comprehended in X,

I_) is comprehended in J,

!N" is comprehended in D J

.*. A is comprehended in X.

Regressive descending.

D comprehends N,

J comprehends D,

X comprehends J j

.'. .X comprehends N.

If now the two figures be considered as put together
on their only common line capable of actual measure
ment, J D in the first figure and D J in the second
(analogue of the middle premise of each sorites), they
will present the following figure :

Fig. 3.

QUESTIONS IN MATHEMATICS.

which may be folded on its interior lines in the form of
a regular tetrahedron, the points N N, N, and X X, X,
respectively, meeting. The two partly dotted lines, D N"
in triangle 2 and J X in triangle 4, will coincide with the
continuous lines N D in triangle 3 and X J in triangle 1
respectively, and the other two partly dotted lines X N
in triangle 2 and N X in triangle 4 (analogues of the
ultimate conclusions of the two sorites), will coincide and
form a continuous line with the arrow heads lying in ad
jacent faces, but pointing in opposite directions.

Thus geometry through its simplest forms makes clear
to the eye as well as to the understanding that its under
lying science — logic — must in its simplest forms exhibit
the relations of three terms, and may extend but is limited
to four, any advance beyond the fourth term being im
possible except by actual investigation, such investiga
tion in the cases represented by the figures (except con
sidered as folded in the form of a solid figure) being also
impossible. The coincidence of the lines of the figures
as above described demonstrates the infallibility of the
logical processes.

The square, in like manner, represents the sorites re
solvable by means of its diagonal into two syllogisms, as

in the folloAving figures :

Fig. 4.

Fig. 5.

QUESTIONS IN MATHEMATICS.

But if the two squares are put together in like man
ner as before, thus :

Fig. 6.

;.\

they cannot be folded on their interior lines so as to form
a solid figure.

Six squares are required to form a cube and three
combinations of two sorites each (corresponding in num
ber to the three regular tetrahedra to which the cube is
equal in volume) are required to describe the surface
instead of one, as in the case of the tetrahedron.

Does not nature clearly point to the regular tetrahe
dron instead of the cube as the simplest unit of measure
of solidity ?

§ 6. The regular triangle as a unit of measure with
an arc of a circle described on any side thereof from the
opposite point as a centre, may be applied to the deter
mination of any plane angle as follows :

Let the angle B D F, in the following figure, be the
angle to be determined, and let the unit of measure,
X J D, with an arc described on the side X J from the
point D as a centre, be applied to it as shown.

QUESTIONS IN MATHEMATICS. 19

Fig. 7.

From the point E, where the line B D intersects the
arc, draw E E' parallel to X D, and measure the line so
drawn. The angle E E ' D will always be an angle of
120° at whatever point in the line D J its vertex may be.
Then, by means of the two known lines DEandEE'
and the known angle E E ' D opposite one of them (D E),
the angle in question may be found. There will be no
ambiguity. The line D E opposite the known angle
E E'D will always be greater than the line E E '.

In like manner, a line may be drawn from the point
E to the line X D parallel to the line D J and measured,
and by means thereof and the known line D E and the
angle of 120° opposite thereto on the line X D, the angle
E D X may be determined.

If the angle to be determined exceed 60° and be less
than 120°, the unit of measure will have to be applied
the second time, and if it exceed 120°, the third time,
and so on until the circuit be completed, the line E E '
being drawn in the last application in which B D will
intersect the side of the unit opposite the point D, and
the sum of the preceding applications being added to
that therein found.

Thus, if the angle to be determined be a right angle,

QUESTIONS IN MATHEMATICS.

the line B D will lie in the second application of the unit,
as in the following figure :

F-ig. 8.

TSs

The point E' will be in the line D X, and the angle
E D E', the same as B D X, will be 'found to be an angle
of 30°, to be added to X D F 60°, as measured by the
first application of the unit.

B D X (30°) + X D F (60°) = B D F (90°).

The triangular unit of measure having described on
one side thereof an arc of its circumscribing circle may
also be applied to the determination of plane angles as
follows :

Let B D F be the angle to be determined, and let the
unit with an arc of its circumscribing circle described on
its side X J be applied to it, as in the following figure :

QUESTIONS IN MATHEMATICS. 21

From the point E, where the line B D intersects the
arc, draw E J and measure the line so drawn. The angle
D E J will always be an angle of 60° at whatever point
in the arc its vertex may be. Then, in like manner as
before, by • means of the two known lines D J and E J
and the known angle D E J opposite one of them (D J),
the angle in question may be ascertained. As before,
there will be no ambiguity. The line D J opposite the
known angle D E J will always be greater than the
line EJ.

If the chord of the arc X E be drawn, the angle D E X
will also be an angle of 60°, and in like manner the angle
E D X may be found.

If the angle to be determined exceed 60°, then, as
before, the unit of measure, if with but one arc described
thereon, as in the figure, will have to be applied the
second time, and if it exceed 120° the third time, and so
on until the circuit be completed.

§ 7. But the circuit can never be completed except
by the continued applications of the unit of measure or
by independent processes of reasoning, for which further
investigation will be required.

This will be manifest by the consideration of the fol
lowing figure, in which the two quadrilaterals, having
their points designated by the logical symbols, are
brought together in such manner that from a common
point designated by the symbol of the third term in each
direction (minor-middle descending and major-middle
ascending) a circle may be described about them in the
circumference of which all the other points will lie.

QUESTIONS IN MATHEMATICS.
Fig, 10.

Let this figure now be compared with one similarly
drawn, but in which the quadrilaterals are squares, as
follows :

Fig. II.

In the former figure, the points designated by the
symbols of all the terms of the sorites have positions
in the circumference except those of the third in each

QUESTIONS IN MATHEMATICS.

process, which are both middle terms, and designate the
centre ; in the latter, only the points designated by the
symbols of the terms of beginning respectively have po
sitions in the circumference with those of the same third
terms designating the centre, and all the points desig
nated by the symbols of all the other terms fall within
the area of the circle.

Which of these figures gives promise, rather than the
other, of simplicity in the operations and processes to be
founded upon them ?

§ 8. A single square having two diagonals, and with
a circle circumscribed may, however, be exhibited, as in
the following figure :

Fig. 12.

All the points, including that of intersection of the
diagonals, are designated by the logical symbols, and
have positions in the circumference and at the centre
of the circle. In order to represent the ultimate points,
N in the descending process and X in the ascending,

24 QUESTIONS IN MATHEMATICS.

as inaccessible, each half of one of the diagonals is ex
hibited in part in two curved lines, one continuous and
the other dotted, the dotted lines being considered as
following the arrow heads introduced.

Here the combination of figures representing the two
sorites is a combination of two right-angled triangles,
each consisting of two smaller triangles, 1 and 2 taken
together and 3 and 4 taken together, instead of a com
bination of quadrilaterals, as in Figs. 10 and 11 ; but
each greater triangle is, in fact, a quadrilateral, the
diagonals each consisting of two lines, forming one and
the same straight line.

Two of the exterior points, it will be seen, are marked
each with two different symbols, viz. : one with J and X,
and the other with D and N, with a line separating the
symbols in each case instead of a space, as in the other
figures. The line X J in triangle 1 is greater than J X
in triangle 4, and N D in triangle 3 is greater than D N in
triangle 2. The inequality of these lines, and generally
the inequality of the lines and angles in this figure as
compared, or rather contrasted, with the equality in
all respects in Fig. 10, besides the confusion arising from
designating two of the points each by two different
symbols must, as it would seem to the author, determine
the question of their likelihood respectively of leading
to simplicity in favor of the latter.-

Again, a square with two diagonals and a circle cir
cumscribed may be exhibited in which the symbols of all
the terms of the sorites shall appear each but once, and
all designating points in the circumference of the circle,,
as in the following figure :

QUESTIONS IN MATHEMATICS.
Fig. 13.

The square in this figure must be considered as con
sisting of two squares, one superposed on the other, and
let it be considered that the one superposed is that con
sisting of triangles 1 and 2, in which the sorites is in
the descending direction, and that the diagonal on the
one beneath consisting of triangles 3 and 4, and in which
the sorites is in the ascending direction, is not drawn
on the one superposed, but shows through the paper.

It will now be seen that the reasoning process in the
ascending direction is in the reverse circular direction of
that in which it has been heretofore exhibited, and that
the two processes run counter to each other, as indicated
by the arrows on the outside of the circle instead of in
the same course, as in all previous illustrations, and as
in like manner therein indicated.

The diagonals are also in opposite directions, and
their apparent point of intersection is undesignated.

The objections in respect to the next preceding figure
as to the inequality of certain specified lines and the
designation of two points, each by two different symbols,
do not apply in this case, but the general objection of in
equality of the lines and angles, as contrasted with the
equality of those in Fig. 10, does apply.

QUESTIONS IN MATHEMATICS.

§ 9. The following figure exhibits a completed circuit
of the square unit of measure, or rather of the right-
angled triangular unit, with a circumscribed circle and
square to which latter the diagonals of the inscribed
square are produced.

Considered as a circuit of the right-angled triangular
unit of measure, it is such with reference to the right-
angled triangles into which the inscribed square is divided
by the diagonals, each of which has its right angle at the
centre of the circle, and not with reference to the right-
angled triangles into which the smaller squares are di-
Tided, none of which has its right angle at the centre,
and considered with reference to the logical processes, it
is founded upon Fig. 12 and not upon Fig. 13, in which the
all-important point, the centre of the circle, is undesig-
nated. The semi-diameters of the circle by which the
inscribed square is divided into four smaller squares
constitute lines of altitude of the right-angled triangular
units produced to the circumference of the circle.

QUESTIONS IN MATHEMATICS. 27

Considered as the completed circuit of the square
unit of measure, it is such with reference to the smaller
squares, and is founded upon Fig. 11.

This may be said to be the primary figure in which
are delineated all the lines constituting the trigonomet
rical functions, those in the figure being the functions of
an arc of 45°, or any multiple thereof by an odd number.

The following figure exhibits a completed circuit of
the regular triangle as the unit of measure, with a cir
cumscribed circle and hexagon, to wilich latter the sides
of the triangles (corresponding to the half -diagonals of
the inscribed square in the preceding figure) are pro
duced, and with lines of altitude of the triangles pro
ceeding from the centre of the circle produced to the cir
cumference in like manner as in the preceding figure.

Fig. 15.

This figure may also be exhibited with additional
lines, as follows:

QUESTIONS IJST MATHEMATICS.
Fig. 15 n

The horizontal diameter of the circle in this phase of
the figure consists of sides (one each) of two triangles
instead of lines of altitude, as in the original phase and
in Fig. 14, and the added lines are those directed to be
drawn in the description of the first method of applying
the triangular unit of measure to the determination of
plane' angles hereinbefore contained. The figures de
scribed by such lines (in respect to each triangle) in con
nection with half the produced sides of the triangles to

•

which they are drawn respectively, are perfect rhombs,
with two angles of 60° each and two of 120° each, and
the lines so drawn are equivalent to the sines and cosines,
as in Fig. 14.

If the lines in this figure, in either phase, were
adopted as trigonometrical functions, each would be
found to have a definite relation to the arc with which
it is connected, which would vary as the arc should
vary similarly to the lines now in use, as in Fig. 14,
and they would be the trigonometrical functions of an

QUESTIONS IN MATHEMATICS. 29

arc of 30°, or of any multiple thereof, by any whole
number.

May it not be that they would lead to greater sim
plicity in the operations and processes to be founded
upon them, than the lines now in use \

The two figures, 14 and 15, are very nearly of the same
degree of complexity, with the difference apparently in
favor of the combination of the circle with squares, but
a comparison of the processes by which they may be
respectively described on paper will show that the differ
ence is in fact in favor of the combination with hexagons.

Thus, with a pair of compasses set at unity and a
parallel ruler, the latter with all the sides of the tri
angles may be described by means of six points marked
by the compasses in the circumference ; but to describe
the former with the diagonals, either the set of the com
passes must be changed after the circle and one diagonal
are drawn, or two other points, besides the six points in
the circumference, must be located without the circle in
order ^ to draw from them the other diagonal, and by
means of the diagonals the inscribed and circumscribed
squares of the circle.

The following figure exhibits the completed circuit of
the triangular unit of measure with an arc of its circum
scribing circle described upon one of its sides (applicable
to the determination of plane angles, as hereinbefore
shown), and having a circle and hexagon circumscribed
with the sides of the triangles and lines of altitude pro
duced to meet them, as in Figs. 14 and 15.

QUESTIONS IN MATHEMATICS.
Fig. 16.

This figure may also be exhibited with additional
lines, as follows :

Fig. 16 «

QUESTIONS IN MATHEMATICS. 31

The arcs described upon the exterior sides of the
regular triangles of which the interior hexagon is com
posed, are equal in length to the arcs of the circum
scribed circle intercepted by the sides of the triangles
produced respectively, and the lines of the circumscribed
hexagon are tangent to each half of both such arcs. The
radius by means of which the arcs are described on the
sides of the triangles is one-half the radius by means of
which the circumscribed circle is described. The cir
cumscribed hexagon is commensurable with the interior
hexagon, the length of the sides of the former being
1,333, that of the latter being 1.

The horizontal diameter of the circle in Fig. IQa
also (as in the case of Fig. I5a) consists of two sides
(one each) of two triangles instead of lines of altitude,
as in the original phase and in Fig. 14, and the added
lines are those directed to be drawn in the description
of the second method of applying the triangular unit of
measure to the determination of plane angles herein
before contained. The hexagon described by all such
added lines taken together is equal to the circumscribed
hexagon in Fig. 15, and each of such lines is a chord of
one-half of the arc of the circumscribing circle of the
original regular triangles, and each two of such lines
forming one and the same straight line is the chord of an
arc of the circumscribed circle in this figure, intercepted
by the lines of altitude of two of such triangles produced.
Such hexagon is incommensurable except in power with
either the interior or circumscribed hexagon.

Might not the study of this figure in both phases lead
to valuable results ?

QUESTIONS IN MATHEMATICS.

But the triangular unit of measure with a complete
circumscribed circle instead of one arc thereof as in Fig. 9,
page 20, may be applied to the determination of all plane
angles as shown in the following figures, in which the
angles to be determined are designated by the letters
B D' F, B' D' F, &c., B D' F being the same as B D F in
Fig. 9, and to be found as described in the text follow
ing that figure.

Fig. 9 a

Bv"

To find B' D' F, as in Fig. 9«, from the point E where
the line B' D' intersects the arc draw and measure E X.
The angle D E X will always be an angle of 120° at what
ever point in the arc its vertex may be. Then, by means
of the known lines D X and E X and the known angle

QUESTIONS IN MATHEMATICS.

D E X, the angle E D X, equal to B' D' X, may be deter
mined, and being added to X D J (60°), will be equal
to B' D' F.

The angles B" D' F in Fig. 9Z> and Bv D' F in Fig. 9c
are measured directly by the unit of measure, and the
other angles in those figures may be found in like man
ner as the angles in Fig. 9a.

The following figure exhibits the triangular unit of
measure with a circle and regular triangle circumscribed
and with lines of altitude produced to the points of the
circumscribed triangle, and chords of all the arcs into
which the circle is divided by such lines of altitude pro
duced.

The sides of the unit of measure furnish the invaria
ble line instead of the radius of the circle as in Figs. 14
to 16a, and the centre of the circle will not lie in the
side of any angle measured, greater or less than 30°.

Would not this figure furnish all the requisite trigo
nometrical functions, and might it not lead to greater
simplicity in the operations and processes to be founded
upon it than the figures before exhibited ?

QUESTIONS IN MATHEMATICS.

§ 10. The oct-astron is the analogue of two independ
ent processes of reasoning, conjoined in the figure, but
in nowise connected, each consisting of two sorites com
bined. In the following figures,

Fig. 17

it is represented in two positions (in each case as seen
with the line of vision perpendicular to the vertical axis
X N' at its middle point, the centre of the figure) ; first,
with triangle 1, as in Fig. 3, on page 16, in full view,
but of the edge of 2, as the tetrahedron was herein first
considered (page 11) ; and, secondly, after having been
turned half-way round on such axis, with triangle a, as
in the next following figure, in full view.

The following figure represents the second of the
intervolved tetrahedra with its faces spread out as a
plane in like manner as in Pig. 3, but differs from
that, not only in respect to the symbols of the terms,

QUESTIONS IN MATHEMATICS. 35

but also in the lateral directions of the two processes of
reasoning, descending to the left instead of to the right,
and ascending to the right instead of to the left. The
tetrahedron formed by the folding of this figure will be
the analogue of the sorites considered as beginning in the
ascending direction, and the tetrahedron, formed by the
folding of Fig. 3, that of the sorites considered as begin
ning in the descending direction, the letters a, 5, c, d
showing the order of the process in this figure, as the
numbers 1, 2, 3, 4 have done in respect to Fig. 3, the tri
angles a and b in the former corresponding to 3 and 4 in
the latter, and c and d to 1 and 2.

(The words "descending" and "ascending" have
been hitherto applied to the processes of reasoning as
exhibited on the faces of a single tetrahedron, descend
ing from X to N and ascending from N to X, in both
cases referring to a single progressive sorites. Let them
be hereinafter considered each as applying to the com-

36 QUESTIONS IN MATHEMATICS.

bined sorites on the faces of each of the tetrahedra inter-
volved in the oct'astron, viz. : descending throughout,
first progressively and then regressively, from X to N, in
the first of such tetrahedra, and ascending throughout,
in like manner, from N' to X' in the second, and let the
expression "complete process of reasoning," when here
inafter employed, signify a combination of two sorites
descending or ascending throughout, unless it shall be
manifest from the context that it is intended to apply
only to one.)

By revolving the oct'astron, as held in the position
shown in Fig. 17, from left to right, triangles 2 and 4, as
in Fig. 3, will successively come in view, and by turning-
it one-fourth of a revolution with its vertical axis as the
diameter of a circle, described by the extremities of such
axis with the point X receding from the eye, triangle 3
will come in full view.

By revolving the oct'astron, as held in the position
shown in Fig. IS, from left to right, triangles b and d, as
in Fig. 19, will successively come in view, and by turn
ing it with the axis as the diameter of a circle in like
manner as before, but with the point N' instead of X
receding, triangle c will come in full view.

It will be seen that there are two axes, the extremities
of which are designated, one by the symbols X W and
the other by N X'. No relation between such symbols
as they are connected by an axis is demonstrated in
either process of reasoning ; but that of X an extremity
of one axis, with N an extremity of the other, and of
N' an extremity of the former with X' an extremity of
the latter. The reasoning is upon lines wholly on the

QUESTIONS IN MATHEMATICS. 37

surface, and not on imaginary lines going through the
body of the figure.

When the entire reasoning process in respect to either
intervolved tetrahedron shall have been gone through
with, there is no going beyond. There is an impassable
gulf between the ultimate point of either intervolved
tetrahedron and the point of beginning in the other which
no process of reasoning unaided by further investigation
can span.

§ 11. The oct'astron has hitherto been considered as
consisting of two intervolved tetrahedra of the edge of 2,
to both of which the included octahedron is common.
Let it now be considered as consisting of the octahedron
as the primary figure, with tetrahedra of the edge of 1
superposed upon its faces, and let each such tetrahedron
be considered as having its points designated on each
face by the logical symbols similarly to the two inter
volved tetrahedra as hereinbefore shown ; that is, four
with the symbols X J D N, and considered as being in
the descending direction throughout, and four with the
symbols N' D' J' X', and considered as being in the as
cending direction throughout ; and let such tetrahedra
be considered as so superposed that the exterior points
of the whole figure shall be designated by the same sym
bols respectively and relatively to each other, as in Figs.
17 and 18, on page 34. Then will the faces of the whole
figure, as brought to view by its revolution, be the same
as in those figures and the description following them,
except that each face will have three small triangles,
with all their points designated, instead of the three

38 QUESTIONS IN MATHEMATICS.

considered as one great triangle, with only its exterior
points designated, as in the figures.

Let it now be further considered that the designations
on the faces of the tetrahedra, which are respectively
applied to the octahedron, and also the designations of
the vertices of such tetrahedra (points of the oct'astron)
opposite such faces are impressed on the faces of the
octahedron, on which such faces of the tetrahedra are
respectively applied ; and let all such tetrahedra be con
sidered as removed.

The faces of the octahedron may then be spread out
in several different ways, in each case in two plane figures,
of which ways two, with the designations, will be as
shown in Figs. 20, 21, 22, and 23, on the next page, to
be taken together as they stand, side by side ; the desig
nations in the second figure in each case being consid
ered as on the other side of the paper.

If the first of these figures in each case be considered
as lifted up to a height equal to the altitude of the
octahedron and placed directly over the second, so
that the centres of the two middle triangles (3 and c
in each case) shall be extremities of a line perpendicular
to the plane of each such triangle, then the exterior
triangles of the upper figure may be folded downward
over the points or sides, as the case may be, by which
they are connected with the middle triangle, and those
of the lower figure upward in like manner, until they
respectively meet, and the resulting figure in each case
will be the octahedron reconstructed. There are no
natural planes passing through the body of the octa
hedron.

QUESTIONS IN MATHEMATICS.
Fig. 20. Fig. 21.

Fig. 23.

Each point of the octahedron will now be found
marked with four different symbols, viz.: two different
ones of each process, as shown in the following illustra
tion, in which two faces only of the octahedron appear,
the lines produced and otherwise exhibited representing

QUESTIONS IN MATHEMATICS.

the edges of the octahedron formed by the sides of the
adjacent and opposite faces, as will be readily under
stood.

No two adjacent faces joined by an edge of the octa
hedron constitute, when spread out, a quadrilateral on
which either process is exhibited as a sorites, but in
every case a syllogism of one of the processes is conjoined
with a syllogism of the other.

If triangle 4 in Fig. 3 (page 16) were turned upward
in a semicircle on the point J as a centre, and if tri
angle d in Fig. 19 (page 35) were turned downward in
like manner on the point D' as a centre, the two result
ing figures would be in the same forms as Figs. 22
and 23, and could be folded and put together in the
form of an octahedron, the points of which would be
found marked each with two different symbols, viz. :
on one face with a symbol of one process and on each
of the other three faces with a symbol of the other
process, as follows :

QUESTIONS IN MATHEMATICS.

.41

••

Two sorites only, one of each process, would be ex
hibited as composed of two syllogisms regularly com
bined on two adjacent faces, viz. : that on faces 1 and 2
taken together, and that on faces a and b taken together.
The syllogism on each of the other faces would not com
bine with that on either of the adjacent faces respectively,
so as to constitute a sorites.

t Figs. 3 and 19 cannot be folded and put together in
the form of an octahedron, but either iigure and a
duplicate thereof may be. The two complete processes
of reasoning in such case would be in opposite circular
directions.

If the two following figures are folded, the first down
ward and the second upward (the symbols in the second
being considered as on the other side of the paper), and
put together so that the edges formed by the meeting of
the lines of the openings in the figures shall be opposite
each other, the two complete processes, but not combined
in regular order as sorites, will occupy four adjacent
faces of the octahedron meeting at a common point.

QUESTIONS IN MATHEMATICS.

Fig. 24,

Fig. 25.

The points of the octahedron will now be found
marked on their four faces respectively, as follows :

This is, in either case, confusion worse confounded.
It is manifest that the octahedron is not the figiire de
signed by nature as the analogue of the perfect and har
monious conjuncture of the two complete reasoning pro
cesses, but instead, that the tetrahedron in which it is
partially concealed is the analogue of one complete pro
cess in respect to four terms (the visible faces of the

QUESTIONS IN MATHEMATICS.

•octahedron in such case having their angles undesig-
nated), and that the oct'astron in which the octahedron
is wholly concealed is the analogue of two complete
processes perfectly conjoined, each showing the relation
of four terms respectively, but each wholly independent
of the other.

. If two octahedra with the logical symbols of both the
complete processes impressed on their faces, one as
secondly and the other as thirdly described, should have
the tetrahedra of either one only of such processes super
posed upon their appropriate faces, the resulting solid
figure in each case, instead of being in the perfect form
of a regular tetrahedron of the edge of 2, as it would be
as first described, would be irregular, as shown in the
following illustrations. The figures are so drawn as to
represent the octahedron in each case with one of its
axes vertical, the figures being considered as held below
the eye. All the symbols except those at the vertices
of the superposed tetrahedra and at the centres of those
of the visible faces of the octahedron which are in sight
in the figures are omitted.

Fig. 26

Fig. 27

44 QUESTIONS IN MATHEMATICS.

The significance of the foregoing description of the
octahedron (in three phases) showing its inadaptability
to be regarded as the analogue of the perfect conjuncture
of the two complete processes of reasoning will not ap
pear until the consideration of the sphere is reached.
The description has been introduced here as in its ap
propriate place following the description of the oct'astron
showing its perfect adaptability.

§ 12. The oct'astron has hitherto been considered as
having one of its axes, X N', vertical and all the others
oblique. Let it now be considered as let fall to one side,
in which position all its axes become oblique. It may
then appear as shown in the following illustration :

Fig. 28

This figure exhibits the oct'astron in the form not of
an outlined but of an out-pointed cube, with intersecting
lines connecting diagonally the points of what would be
the faces of the cube if it were outlined. The figure
may, perhaps, properly be called the skeleton of a
cube.

QUESTIONS IN MATHEMATICS. 45

Let the figure now be considered as divided by three
planes passing through the centre of the included octa
hedron in line with its three axes and edges, and there
will result eight equal figures, each consisting of a regu
lar tetrahedron, with an irregular right-angled tetrahe
dron attached to one of its faces.

Twenty-four imaginary right-angled tetrahedra must
now be supplied, three to be applied to the faces of the
tetrahedron in each of such eight figures, and the figures
considered as put together again before a perfect cube of
the apparent dimensions of 2, in length, breadth, and
height, can be imagined. Such dimensions will not be 2,
but V"2 = 1.4142. But the diagonals of the faces will
each be 2.

It will thus be seen that, while a regular tetrahedron
of the edge of 1 is the nucleus of a cube of the edge of
.7071, its edges being diagonals (one of each) of the faces
of the cube, as hereinbefore shown, a regular octahedron
of like edge (1) is the nucleus of a cube of the edge of
1.4142, its points being the points of intersection of both
diagonals of the faces of the cube, and that a regular
oct'astron of like edge (1) is also such nucleus, its edges
being both diagonals of the faces, and its points the
points of the cube ; the tetrahedron, the octahedron, and
the oct'astron being otherwise than as described wholly
hidden within the body of the cube.

If the faces of the cube having its points designated
by the logical symbols, as such points are designated on
the faces of the included oct'astron as in Figs. 17 and 18,
on page 34, be spread out, such designations will be found
to be as shown in the following illustration :

QUESTIONS IN MATHEMATICS.
Fig. 29

D'D

Here again is confusion, even worse confounded than
in the case of the octahedron. Two symbols of each
process appear on each face, but in no case, considering
the terms of the two processes as interchangeable but
as retaining their logical significations, can a sorites be
found. Two of the faces have each two magnus terms,
but no maximus term ; two have each two maximus
terms, but no magnus term ; and the remaining two have
each, the magnus and maximus terms, and the two mid
dle terms diagonally opposite respectively, instead of
being, the two former, extremes of one of the sides, and
the two latter, extremes of the opposite side.

QUESTIONS IN MATHEMATICS. 47

The oct'astron included in the cube as represented in
the foregoing figure is considered as consisting of two
intervolved tetrahedra, and if diagonals had been drawn
on each face of the cube, their point of intersection
would have been undesignated. But if the included
oct'astron is considered as consisting of tetrahedra super
posed on the faces of its included octahedron as described
in § 11, on page 37, then it would be necessary to draw
such diagonals and their point of intersection on each face
would be found designated by eight symbols (different
on different faces), two in each of the four triangles into
which the face would be divided by such diagonals. Such
symbols in the upper face of the cube would be (begin
ning with the triangle at the left hand and going from left
to right) XX', X'D, DD', and D'X. The confusion in
such case would seem to be inextricable. It will be here
inafter shown that the oct'astron must be considered as
consisting of tetrahedra both intervolved and superposed.

But the faces of the cube may have three combina
tions of two sorites each, in regular form, as shown in
Fig. 30 (next page), in which let the first sorites of each
combination be considered as in the descending direction
and the second in the ascending (instead of being de
scending or ascending throughout), each combination
beginning at a different point from either of the others,
but all terminating at the same point. If a regular
tetrahedron be considered as the nucleus of the cube,
then assuming the diagonals of the three faces on which
are given the three sorites in the descending direction
to be thrfte edges of such tetrahedron, the diagonals of
the other three faces are not the other three edges.

QUESTIONS IN MATHEMATICS.
Fig. 30

The following figure

Fig. 31

\ 1

\
\

TV/

\ 2

\
\

QUESTIONS IN MATHEMATICS.

is the same as the preceding in respect to such iirst-men-
tioned three faces, but in respect to the other three the
diagonals are in the opposite directions respectively, and
are edges of the included tetrahedron, but the sorites in
the ascending direction cannot be represented upon such
faces respectively, except by transposing the terms as
they appear in Fig. 30, in which case, the two sorites of
each combination would be found to run counter to each
other, in opposite circular directions.

In neither of the combinations shown in Fig. 30 does
the second sorites return to the point of beginning of the
first, as in the case of the combination on the faces of
the tetrahedron. But they may be exhibited in such
manner, as in the following illustration,

Fig. 32

•

that in each case the second sorites shall return to the

50 QUESTIONS IN MATHEMATICS.

point of beginning of the first, but such point will not be
the same in respect to any two combinations.

In the foregoing figure it will be seen that each of the
combinations has the line X N or IN" X (analogue of the
ultimate conclusion of each sorites) common, instead of
the line J D or D J (analogue of the middle premise in
each sorites), which wras the only common line capable of
actual measurement in the original construction and com
bination of the figures.

In like manner as before, three only of the diag
onals are lines of edges of the included regular tetra
hedron.

To return to the oct'astron. In one or the other of
the two aspects in which it has been exhibited — that is,
either first as consisting of a combination of two inter-
volved tetrahedra or of eight tetrahedra superposed upon
the faces of an octrahedron, or secondly as the nucleus of
a cube (its dissection as described in § 4 being taken in
connection with the second) — must the oct'astron be re
garded in order to find a unit upon which to base the
operations and processes of geometry looking to the
measurement of its volume. In the first, the analysis is
along natural lines and actually existing planes lying
wholly on the surface (each tetrahedron, either inter-
volved or superposed, being considered by itself) ; in the
second, with the exception of the diagonals, it is wholly
along imaginary lines and planes, some of which lie
wholly on the imagined surface, and the others wholly
within the body of the figure.

Which of the two aspects is the simpler and the more
in accord with nature ?

QUESTIONS IN MATHEMATICS. 51

§ 13. Twelve octahedra (edge 1) may be superposed
upon the twenty-four faces of the oct'astron, each octahe
dron superposed upon or perhaps rather interposed be
tween two adjacent faces, one each of two of the tetra-
hedra (edge, 1) of the oct'astron, and twenty-four tetra-
hedra may be interposed, each between two adjacent
faces, one each of two of such octahedra, one edge of
each of such interposed tetrahedra falling, upon and
coinciding with an edge of one of the tetrahedra of
the oct'astron, and six octahedra may be interposed,
each between four faces, one each of four of such in
terposed tetrahedra meeting at a common point, and the
resulting figure will be a regular octahedron of the
edge of 3. One point of the included and wholly hid
den oct'astron will be at the centre of each face of such
octahedron.

It is now manifest that such octahedron will be the
nucleus of a second oct'astron of the edge of 3, composed
of two intervolved tetrahedra of the edge of 6, and that
such second oct'astron will be the nucleus of a third octa
hedron of the edge of 9, which will be the nucleus of a
third oct'astron of the edge of 9, composed of two inter
volved tetrahedra of the edge of 18, and so on, ad in-
finitum.

The first octahedron (the central figure of the first
oct'astron) may be called an octahedron of the first
order, and the tetrahedron and oct'astron formed thereon
a tetrahedron and an oct'astron of the first order, and
the second of each, of the second order, and so on.

The primary figure is evidently the octahedron on
which both the others are constructed.

52 QUESTIONS f IN MATHEMATICS.

The following table exhibits the edge and volume of
each figure up to and including the sixth order :

TABLE OF EDGES AND VOLUMES

OP THE OCTAHEDRON, TETRAHEDRON, AND OCT'ASTRON, BEGINNING
WITH UNITY AS THE EDGE OP THE OCTAHEDRON.

ORDER.

OCTAHEDRON.

TETRAHEDRON.

OCT'ASTKON.

Edge

Vol.

Edge

Vol.

108

216

324

Edge

Vol.

2916

5832

8748

Edge

Vol.

78732

157464

236196

Edge

162

Vol.

2125764

4251528

6377292

Edge

243

486

243

Vol.

57395628

114791256

172186884

The edge of the tetrahedron of each order is in all
cases double that of the octahedron and oct'astron of
the same order.

The volume of the tetrahedron of each order is double
and that of the oct'astron three times that of the octahe
dron of the same order.

QUESTIONS IN MATHEMATICS.

The edge of each figure of each order after the first is
3 times and the volume 27 times that of the same figure
of the next preceding order.

The volume of the octahedron of each order after the
first is 13£ times that of the tetrahedron and 9 times that
of the oct'astron of the next preceding order; and the
volume of the tetrahedron of each order after the first is
54 times and that of the oct'astron 81 times that of the
octahedron of the next preceding order.

To find the edge and volume of the octahedron of any
order, take the number of the next preceding order as
the exponent of a power, and raise 3 and 27 to such
power. The power of 3 will be the edge, and that of 27
multiplied by 4 will be the volume required.

Thus, the edge and volume of each figure of the tenth
order are as follows :

ORDER.

Edge

OCTAHEDRON.

TETRAHEDRON.

OCT ' ASTBON.

2(3')

Vol.

4(279)

8(279)

12(279)

The volume of each figure of any order being 27 times
that of the same figure of the next preceding order, the
difference between the volumes in any two consecutive
orders will, of course, be the product of the volume in
the first of such orders multiplied by 26. Such differ
ence, except in the case of the oct'astron, is also equal to
the product of 26 multiplied by twice the difference be
tween the numbers of the edges of the same figure of the
two orders, and again by 9 raised to a power, the expo-

QUESTIONS IN MATHEMATICS.

nent of which is equal to the number of the first of such
orders — 1.

Thus, such differences in respect to the octahedron
and tetrahedron up to the sixth order are as follows :

DIFFERENCES BETWEEN VOLUMES.

Octahedra.

Tetrahedra.

1st

and 2d

26 X 4 X 0°

26 x 8 x 9°

" 3d

26 x 12 x 91 26 x 24 x 91

" 4th

26 x 36 x 92

26 x 72 x 92

4th

" 5th

26 x 108 x 93

26 x 216 x 93

5th

" 6th

26 x 324 x 94 26 x 648 x 94

and so on.

To apply the above process to the oct'astron, it is
necessary to multiply the difference between the num
bers of the edges given in the table also by 3.

The author confesses himself to have been and still
to be in a quandary as to whether the oct'astron of the
first order should be described as of the edge of 2, being
that of each intervolved- tetrahedron, or of 1 being that
of each superposed tetrahedron. As some description
seemed to be necessary, the latter has been adopted. It
is manifest that it cannot be regarded as of the edge of 3.

The tetrahedron of the first order in the table is of
the edge of 2. Let it now be considered that it is in
fact a tetrahedral yard, each edge being one yard in
length. The edge of each smaller tetrahedron, of four
of which it in part consists, will then be half a yard or

QUESTIONS IN MATHEMATICS. 55

one and a half feet, and let it be considered that it is de
sired to ascertain the contents of each of the three figures
of the first order, and then of the second and succeeding
orders in the table in terms of tetrahedral feet.

The edges of the figures of the first order in such
terms will be 1.5, 3, and 1.5 respectively. To ascertain
the volumes find the third power of 1.5 = 3.375, and
multiply such third power by the volumes in the first
order, as in the table, viz.: 4, 8, 12. To find the edges
and volumes in the second order multiply the edges in
the first order as above by 3, and the volumes as so found
by 27, and proceed in like manner to find the edges and
volumes in the third order, and so on.

Similarly the volumes could be found with 2, 2.5, or
any number, whole or fractional, as the edge of the octa
hedron of the first order.

If a table were constructed upon any such octahedron
of greater or less edge than 1, as of the first order, the
edges and volumes of the figures of all the orders would
have the same definite relations to the edges and vol
umes of the figures of the corresponding orders in the
foregoing table throughout, as in the first order. No
such table would, therefore, be required.

In all cases where the edge of the tetrahedron is an
odd number, procedure in physical construction upon
the octahedron of edge 1 as the central figure would be
upon artificial lines and faces produced by forced sections
of the regular figures. While this is practicable by
means of fractions in arithmetic, which has to do only
with abstract numbers, it would be utterly impracticable
in physical geometry.

56 QUESTIONS IX MATHEMATICS.

In what way the building up of the ligures and the
table may serve in geometrical processes the author is
unable to say, but he will hereinafter show that the tetra-
hedra in an octahedron of the second order are analogues
of compound logical processes through which the two
complete simple processes on the faces of the ocfastron
of the first order are brought into perfect union. The
table will be herein called the table of natural involution.

If a regular tetrahedron (edge 1) be taken as the cen
tral figure, and be built upon by superposing octahedra
upon its faces, and interposing tetrahedra between the
faces of such octahedra, the resulting figure will be found
to be an irregular octahedron having four of its faces
regular hexagons of side \ (a point of the central figure
being at the centre of each of such faces), and the other
four, regular triangles of side 1. By superposing regular
tetrahedra upon the latter four, the further resulting
figure will be a regular tetrahedron of edge 3.

This figure could be again in like manner built upon,
and the second ultimately resulting figure would be a
regular tetrahedron of edge 9, and so on. But the first
resulting figure would in all cases be irregular.

1 If a table were constructed upon such a series it
would be one of tetrahedra only, the edge of which in
each order would be one-half and the volume one-eighth
those of tetrahedra of the corresponding orders in the
table of natural involution. Such table would therefore
not be required.

§ 14. A single regular tetrahedron, the unit of meas
ure, may be divided by four sections into five fractional

QUESTIONS IN MATHEMATICS. 57

parts, of which four will be each the one-eighth part,
each in the form of a regular tetrahedron of edge £ and
volume |, and the fifth, the one-half part, a regular
octahedron of the same edge, and equal in volume to the
other four parts combined.

This octahedron may now be subdivided by dissection
into fifty-one parts (equal to the number of figures in an
octahedron of the second order), of which, nineteen will
be regular octahedra (one of which will be the central
figure), and thirty-two regular tetrahedra. The edge of
each such tetrahedron will be |-, and the volume ^\$ of
the edge and volume of the original unit.

In like manner the central octahedron may be sub
divided, and the edge of each smaller tetrahedron will
be ^, and the volume ^^2 °f the edge and volume of the
original unit, and so on.

If the table of natural involution be considered as
extended in the opposite direction from the point of
beginning, in respect to the tetrahedron only, beginning
with a regular tetrahedron of edge 1 divided as above de
scribed, and the orders correspondingly numbered back
ward (or more properly in the descending direction), the
edge and volume of the tetrahedron of each order will
be reciprocals respectively of the edge and volume of
the tetrahedron of the order of corresponding number in
the forward (ascending) direction.

The centre, it is obvious, can never be reached in the
descending direction, how far soever the process may be
continued. The central figure will always be that of an
octahedron. If it be attempted to divide the octahedron
in anv other manner so as to reach the centre, the re-

58 QUESTIONS IN MATHEMATICS.

suiting figures will be irregular, and irregularity in figure
must, as it would seem to the author, necessarily involve
intricacy in calculation.

Does not this, in connection with what has been here
inbefore shown as to the perfect accord between the
logical and geometrical processes along the lines of the
faces of the tetrahedron, seem to make manifest that
nature forbids any attempt to reach the centre of a
solid figure, and that all processes relating thereto should
be conducted along natural lines and planes lying on
surfaces as originally existing or as superposed, or
by natural sections disclosed, and not along lines and
planes produced by forced sections seeking to reach the
centre ?

If a regular tetrahedron of edge 1, the unit of meas
ure, be considered as divided in the reverse direction of
the process of building upon it as the central figure,
as before described, such division would necessarily
begin with cutting off from each point a regular tetra
hedron of edge J and volume ^V, leaving an irregular
octahedron for further division, as described in the
building-up process. The tetrahedron to be cut off in
the second instance would be of edge ^ and volume ?^g,
and so on.

If the table suggested in connection with the descrip
tion of such building-up process (but which would never
be required) be considered as extended in the opposite
direction, in like manner as described in respect to the
extension of the table of natural involution, the edge and
volume of the unit would constitute the first order in

QUESTIONS IIST MATHEMATICS. 59

each direction, and the edge and volume in any order in
either direction would, in like manner, be reciprocals re
spectively of the edge and volume in the order of cor
responding number in the other direction.

This, in respect to the edge and volume of the unit
constituting the first order in the descending direction,
may at first seem paradoxical, but a comparison of the
two tables will show that it is true, and further, that it
is requisite in order that the relations of the two tables
to each other in both directions, and of each order in
either table in one direction to the corresponding order
in the same table in the other direction, may be sym
metrical throughout.

In the description of the suggested table (based upon
the tetrahedron as the central figure) in the ascending-
direction, it was stated that the edge of the tetrahedron
of each order therein would be one-half and the volume
one-eighth that of the tetrahedron of the corresponding-
order in the table of natural involution.

This is exactly reversed in the extension of the two
tables in the descending direction. The edge of the
tetrahedron of every order, beginning with the first in
the descending direction of the suggested table is twice
and the volume eight times the edge and volume re
spectively of the tetrahedron of the corresponding order
in the descending direction of the table of natural in
volution.

The two tables in both directions up to and in
cluding the fourth order, are as follows, the table of
natural involution being exhibited only as to the tetra
hedron :

QUESTIONS IN MATHEMATICS.

TABLE

2n TABLE

OF NATURAL INVOLUTION.

AS SUGGESTED.

Of the Tetrahedron only.
ORDER.

Of the Tetrahedron.

Edge. Volume.

Edge.

Volume.

^ f 4 54 157404

19083

1 3 18 t

5832

729

1 2 6

210

3 •

1 2

b r i s-

i
i

i J °
1 \ 3 iV
. ^ -h

2T3"

Tlir
Tfflhrff

The tetrahedron is the principal figure in the table of
natural involution, the octahedron being the primary
figure npon which it is constructed, and the oct'astron,
as it were, a compound tetrahedron, wholly lost sight of
in the descending direction of the table.

The unit 1 has no place in the table, considered with
reference to the tetrahedron only, but is the base of the
first order in both directions, in which the edges are both
its product and quotient by 2, and in the full table it
has place only as the representative of a line which is no
real but simply an ideal thing.

If the orders of the decimal scale had been in like
manner numbered, with the unit 1 regarded as of the
first order in both directions, each subsequent order of
corresponding number in both directions would have
been reciprocals of each other. Thus,

QUESTIONS IN MATHEMATICS.

DECIMAL SCALE

(Aa suggested).

ASCENDING.

DESCENDING.

5 o

1st Order

2 v ~

o o

in both

1 • 1 5

& —

Directions.

— i "B £

"**

N CO T^

•o

100000

10000 1000

100

.1 .01 .001

.0001

.00001

That the first order should occupy but one place as
the base in both directions instead of two, as in the tables
of tetrahedra above given, is, as it would seem to the
author, fairly to be inferred upon consideration of the
following diagram in the form of a regular triangle in
which, with the unit at the vertex, the first terms of the
successive orders of difference in both directions are
shown to be connected laterally by a regular gradation
of intermediate differences.

\ \ \ v

10000\ 1000 \ 100 \ 10

.1 / .01 / .001 /flora

62 QUESTIONS IN MATHEMATICS.

Is not the difference between the tables and the scale
in this respect to be accounted for by the fact, that in
the two former the thing considered as the base in each
direction is concrete (whether regarded either as an actual
solid or simply as a volume of space), but in the latter is
abstract ?

From the foregoing considerations it would seem that
the unit 1 is simply the base of all numbers and quanti
ties, by means of which they are measured, and cannot
itself be regarded as in any sense a number.

The central figure of any course of a regular tetrahe
dron, the number of which is 3 or 3 + 1, or any multiple
of 3, or any such multiple + 1, is a tetrahedron, and that
of any course the number of which is 3 — 1, or any mul
tiple of 3, — 1, is an octahedron.

The central figure of a right triangular pyramid the
number of courses of which is any odd number, is a regu
lar tetrahedron, and that of one the number of courses
of which is 2 or the sum of 2 and 4, or of 2 and any mul
tiple of 4, is a regular octahedron, and the centre of such
figure in either case is the centre of the pyramid. But if
the number of courses is 4 or any multiple of 4, there is
no central figure, and the, centre of the pyramid in such
case is at the point of intersection of four planes passing
through the pyramid parallel to and equidistant from its
four faces respectively, and each between two courses
relatively to one of the faces and its opposite point.
Such point of intersection is common to fourteen figures
(edge 1) viz. : six octahedra and eight tetrahedra, which
together constitute a regular octahedron of edge 2 and
volume 32. Such edge is twice and volume eight times

QUESTIONS IN MATHEMATICS.

the edge and volume of the octahedron of the first order
in the table of natural involution.

Thus it appears that in such case nature does not for
bid an attempt to reach the centre. But she permits
it only in the ascending direction, and in cases only
in which it can be accomplished by a process regular
throughout.

The fo rejoin o1 obser-

OWA11O *•

vations will more clearly
appear by the accompa
nying diagram, showing
a section of the central
figures in the several
courses made by a plane
passed perpendicularly
from the vertex to the
base of a pyramid of
edge 8, and considered
also as a series of pyra
mids of edges from 1 up
to 8, the arrows pointing
to the centre of a pyra
mid of each edge con
secutively. The centre
of the pyramid is at a
point in its line of alti
tude three-fourths the
length thereof from the
vertex and one-fourth from the base.

The number of the course in which, or of the first of
the two courses in the plane dividing which, the centre

64 QUESTIONS IN MATHEMATICS.

of a pyramid of any given edge is to be found may be
ascertained by subtracting from the number, of the
given edge the whole number contained in the quotient
arising from the division of the number of the given
edge by 4. The remainder will be the number of the
course required.

Thus the centre of each, successively, of a series of
pyramids of edges as follows will be

Of edge 159 in a tetrahedron in course 120,
Of edge 160 between courses 120 and 121,
Of edge 161 in a tetrahedron in course 121,
Of edge 162 in an octahedron in course 122,
Of edge 163 in a tetrahedron in course 123,
Of edge 164 between courses 123 and 124,
Of edge 165 in a tetrahedron in course 124,

and so on.

§ 15. The powers of numbers were undoubtedly first
derived from the consideration of a square and cube
divided into smaller equal squares and cubes, of which
one in each case was regarded as the unit of measure,
and hence the name of square for the second power
and of cube for the third. All higher powers are mere
multiples.

Such powers, considered as involving the notion of
space, are the same and calculable in like manner in
the case of the regular triangle divided into smaller equal
regular triangles, as in the case of the square ; and in
the case of the regular tetrahedron, considered as divided

QUESTIONS IN MATHEMATICS. 65

into smaller equal volumes of regular tetrahedra, as in
the case of the cube.

This will be evident as to the regular triangle upon
mere inspection of such a triangle so divided. But it
will not be so in the case of the regular tetrahedron.

Let a regular tetrahedron of the edge of 1, the unit
of measure, be regarded as standing, not upon either face
as a base, but upon one of its points with the opposite
upturned face horizontal.

This will be the first course of a regular tetrahedron
or inverted right triangular pyramid of any number of
uniformly increasing courses to be superposed thereon
successively.

In constructing the second course, the first figure will
obviously be an octahedron to be superposed on the up
turned face of the single tetrahedron of the first course.
On the three lateral faces of the octahedron pointing
downward let there be superposed three tetrahedra and
the second course will be complete. Represented by
numbers, it consists of the octahedron equal to 4 tetra
hedra, and the 3 superposed tetrahedra, making the
volume of the second course equal to 7 tetrahedra,
which, added to the number in the first course, 1, makes
8 = 2x2x2.

Thus far all seems simple enough, but when the next
course is considered there will be found a unit which has
no visible representation on the external face. The fol
lowing figure exhibits such external face, each smaller
triangle being marked T or O, to signify that it is the
face of T, a tetrahedron, or O, an octahedron.

66 QUESTIONS IN MATHEMATICS.

Fig. 33.

Here are three octahedra and six tetrahedra. The
volume of the course, as calculated from an external
view, would be 3 x 4 + 6 = 18, to which add the volume
of the two preceding courses, 8, and the total apparent
volume of the pyramid of three courses is 26.

But by examining the other side of the third course,
the lateral faces (one of each) of the three octahedra
meeting at their common point in the centre of the ex
ternal face, as above shown, will be found diverging from
such point along the three coinciding lateral edges of such
octahedra until they meet and their lower horizontal
sides coincide with the sides of the upper horizontal face
of the single octahedron in the preceding course, making
a volume of space equal to the unit of measure. This
unit added to the volume found as above, makes the total
actual volume of the tetrahedron whose edge is 3, 27 —
3 x-3 x 3.

In the fourth course the number of such concealed
units will be 3, in the fifth 6, in the sixth 10, and so onr

QUESTIONS IN MATHEMATICS. 67

in a series, the differences of the terms of which increase
by 1 as each course is successively superposed.

The volumes of the several courses constitute a series,
the first order of differences of which begins with 6 and
increases in multiples thereof by 2, 3, 4, and so on, the
second order of differences being, of course, in each
case 6. Thus,

Series. 1, 7, 19, 37, 61, 01, &C.

1st Order of Differences. 6, 12, 18, 24, 30, &Q.

2d Order of Differences. 6, 6, 6, 6, &C.

Such series cannot be found in the courses of a cube
all of which, considered with reference to any one side
as a base, are equal to each other, except by considering
and taking together for each term of the series consecu
tively the whole course on one side of the cube, a part
of the corresponding course on a second side, and a
still less part of the corresponding course on a third
side, the three sides having a common point. But
such series, omitting the first term, is the first order of
differences of a series of volumes of cubes beginning
with 1.

§ 16. The following table exhibits an analysis of the
several courses of a regular tetrahedron of the edge of
12, and also of a series of regular tetrahedra (whole
figures) of edges from 1 up to 12.

QUESTIONS

MATHEMATICS.

ANALYSIS OF REGULAR TETRAHEDRON.

No. OF COURSE AND
EDGE OF FIGURE.

OF COURSES.

OF WHOLE FIGURES.

I i ! i

a .2
S «i _: • 3

o c
HO > EH

fi d

rQ *H

.§ O
jg

CJ V

EH O

"S ^4

t-> o
c >

5 1

l + ( 0x4= 0)= 1

l+( 0x4=

0)= 1

3+( 1x4= 4)= 7

4+( 1x4=

4)= 8

7 + ( 3x4= 12)= 19

ll + ( 4x4=

16)= 27

13 + ( 6x4= 24)= 37

24 + ( 10x4=

40)= 64

21 + (10x4= 40)= 61

45 + ( 20x4=

80)= 125

31 + (15x4= 60)= 91

76 + ( 35x4 =

140)= 216

43 + (21x4= 84) = 127

119 + ( 56x4 =

224)= 343

57 + (28x4 = l]2) = 169

176 + ( 84x4=

336)= 512

73 + (36x4 = 144) = 217

249 +(120x4=

480)= 729

91 + (45x4=180) = 271

340 + (165x4 =

660) = 1000

111 + (55x4=220) = 331

451 + (220x4 =

880) = 1331

133 + (66x4= 264) = 397

584+(286x4 = 1144) = 1728

Let n signify the number of any course or the edge of
any whole figure, s the solidity of any course, and S the
solidity of any whole figure.

The number of tetrahedra in any course is equal to

n (n — 1) + 1, and also to

(n - I)2 + (n + I)2

The number of octahedra in any course is equal to

-, and is also equal to one-half the number less 1

of tetrahedra in the same course.

QUESTIONS IN MATHEMATICS. 69

The volume of the octahedra in any course is equal
to double the number of tetrahedra in the same course,

less 2.

The volume of any course is equal to 3 times the
number of tetrahedra in the same course, - - 2. It is
also equal to 6 times the number of octahedra in the
same course, + 1. The expression in either case may
be reduced to

s = 3n(n — 1) -f 1.

It is also equal to twice the sum of the numbers of
both figures in the same course, — 1.

To find the numbers of the tetrahedra and octahedra
of which a regular tetrahedron of any given edge (??) or
of any given volume (S) consists, it is necessary to begin
with the octahedra.

The number of octahedra in a regular tetrahedron of
any given edge is equal to ril x - — , and of any given

volume is equal to - — , but n in the second expres
sion is an unknown quantity.

The number of tetrahedra in a regular tetrahedron of

(71 Tl\
11? x — — — 1 + n, and of

any given volume is equal to 2 (— - — — ) + n, but n in

\ b by

the second expression is an unknown quantity.

The volume of any regular tetrahedron is equal to

70 QUESTIONS IIST MATHEMATICS.

3 times the number of the tetrahedra therein contained,
- 2ft, and also to 6 times the number of the octahedra
therein contained, + n.

It may be expressed in either case thus :

It is also equal to twice the sum of the numbers of
both figures therein contained, — n.

It is also equal to the product of the number of tetra
hedra in the n^ course multiplied by (n + 1), — 1.

It is also equal to- the product of the number of octa

hedra in the wth course multiplied by 4 i -- + -- 1 + n.

\ a ii I

mu f n . »/ S

The value of - is equal to \ / ^—

6 V 6° = $

216"

The value of n, when the volume of a regular tetrahe
dron only is given, may be found by the usual arithmetical
process <~ *' extracting the third root, or by a very much
simpler xd, with the aid of the following table, in

which (L &£*, " as filled up and made complete) is
shown the nu ^ "tahedra in every regular tetrahe
dron, the quotien. ft 1o>e °f which, divided by 6, is
a whole number from ^ *O V "7.

Let E signify the nuniu ^O/v qdge or root in the
last column, such number as eu '/fa "" the process to
be shown being approximate, n beiii^ /^ bol of the
required edge or root when found.

The numbers shown in the first column of differences
are negative quantities, and those in the second column,
positive.

QUESTIONS IN MATHEMATICS.

TABLE

OF OCTAHEDRA (EDGE 1) CONTAINED IN REGULAR TETRAHEDRA OF
EDGE GIVEN IN LAST COLUMN.

VALUE OP
n E
6 °r T

NO. OP OCTAHEDRA.

EDGE OB
DIFFERENCES BETWBEN Nos. OCT. IN BOOT.
KEGULAH TKTKAHEDRA OP EDGES. ,

known.
E and E — 1. E and E + 1. E approxim'te

- 15

286

— 66

969

- 153

171

2300

-276

300

4495

-435

465

30-

7770

- 630

666

12341

- 861

903

4-2

18424

- 1128

1176

26235

- 1431

1485

35990

- 1770

1830

47905

- 2145

2211

62196

-2556

2628

72 '

* *

=* *

34930665

— 176121

176715

594

100

35999900

— 179700

180300

600

101

37090735

- 183315

183921

606

* * *

* *

274 740549390

— 1350546

1352190

1644

* * * * * *

* * *

793 17952380459

- 11316903 11321661 4758

* * * *

* * * * #

1216 64729643840

— 26612160 26619456 7296

* * * *

* * * * *

1667 ' 166766685001

- 50015001 50025003 10002

72 QUESTIONS IN MATHEMATICS.

To find, by means of the foregoing table, the edge of
any regular tetrahedron of which the volume only is
given, or to find the third root of the greatest third power
and remainder over, if any, contained in any given num
ber, divide the given number by 6, find in the table the
number of octahedra nearest the quotient and subtract
such number from the quotient. The remainder, when
the nearest number exceeds the quotient, will be a nega
tive quantity. If there are two numbers equally near the
quotient, either may be taken. This can never occur
when the given number is an exact third power.

Note the number of the edge in the same line with
the nearest number taken.

Observe now whether the remainder exceeds, if it be
a positive quantity, the number shown in the second
column of differences in the table, or, if it be a negative
quantity, that shown in the first column, and if it does,
then subtract therefrom, if it be positive, the difference
between the numbers of octahedra in E and E + 1, being
the number shown in the second column of differences in
the table ; but if it be negative, then the difference be
tween the numbers of octahedra, in E and E — 1, being
the number shown in the first column of differences, and
to be considered as a negative quantity, and continue
subtraction successively, if necessary, namely, in the first
case, of differences between the numbers of octahedra in
E + 1 and E + 2, and between those in E + 2 and E -f 3
(not shown in the table but readily found), or in the second
case, of differences between the numbers of octahedra in
E — I and E — 2 and between those in E — 2 and E — 3
(also not shown in the table but readily found), until the

QUESTIONS I1ST MATHEMATICS. . 73

remainder in the first case shall be less than the next
difference, or, in the second case, shall become a positive
quantity, equal to or greater than the quotient of the
number of the least edge which shall have come into the
process divided by 6. Then, from such remainder, sub
tract the .quotient by 6 of the number of the greatest
edge which shall have come into the process (that is?

E + 3 or 2 or 1\ .

— ^ — — in the first case, or 01 the least edge

, . E — 3 or 2 or 1\ . ,
that is, - in the second case.

\ o /

The number of the greatest edge in the first case or
least in the second, viz., E + 3 or 2 or 1, or E — 3 or 2 or 1,
as the case may be, will be the required root. If there
be no remainder, the given number will be a perfect third
power ; but if there be, then multiply the remainder by
6, and the product will be the excess of the given num
ber over and above the greatest third power therein con
tained, and will be the last remainder that would be
found in the usual arithmetical process of extracting the
third root.

If the first remainder, namely, that arising from the
subtraction of the nearest number from the quotient, does
not exceed the difference in either case, as hereinbefore
directed to be observed, then E will be the required root,

provided such remainder be equal to or exceed — , and

the given number will be a perfect third power if it be

E
equal ; but if it exceed, then subtract therefrom --- and

multiply the remainder by 6, and the product will be

74 QUESTIONS IN MATHEMATICS.

the excess of the given number over the greatest third

power therein contained.

•p<
If the first remainder be less than — , then E — 1 will

be the required root, and the process is to be continued
in like manner as before described in the case where the
first remainder is a negative quantity, although such
remainder in this case will not be a negative quantity
unless the given number be less than E3 — E. The point
at which the first remainder changes from one kind of
quantity, positive or negative, to the other, or rather be
comes 0, is that where the given number is E3 — E.
The point at which subtraction from the first remainder
begins with a positive quantity, and is continued there
after with positive quantities on the one side, and on the
other begins with a negative quantity, and is continued
thereafter on the same side with negative quantities, is
not that at which the first remainder changes from one
kind of quantity, positive or negative, to the other, but
that where the given number is E\

To illustrate. Let the given volume of a regular tetra
hedron of which the edge is required be . . 729

Divide it by 6. Quotient 121.5

Find and subtract nearest number of octahe-

dra in table, noting edge (E . 6) . . . 35

First remainder 86.5

Subtract difference between numbers of octa-
hedra in E (6) and E + 1 (7), as shown in
table 21

Remainder 65.5

QUESTIONS IN MATHEMATICS. 75

Subtract diff. bet. nos. oct. in E + 1 (7) and
E' + 2 (8), to be found as follows :

Difference taken from table as above 21

+ E + 1 = 7 28

Remainder 37.5
Subtract diff. bet. nos. oct. in E + 2 (8) and

E + 3 (9) = 28 + 8 = 36

Remainder 1.5
Remainder being positive, and being now less

than next difference, subtract - 1.5

6 o

There being no remainder, the given number is a per
fect third power of which 9 (the greatest edge which has
come into the process) is the required root.

Let it be required to find the greatest third power
and remainder over, if any, contained in the given num
ber 963, and also the root of such power.

Given number 963

Divide it by 6. Quotient 160.5

There are two numbers of octahedra in the
table which are equally near the quotient, viz.:
35 (E . 6) and 286 (E . 12). Let the less be taken
as the nearest number.

Subtract nearest no. (E . 6) . . . .35

First remainder 125.5
Subtract diff. bet. nos. oct. in E (6) and

E -f 1 (7), as in table 21

Remainder 104.5

76 QUESTIONS IN MATHEMATICS.

Subtract diff. bet. nos. oct. in E + 1 (7) and

E + 2 (8) = 21 + 7 = . . ' . 28

Remainder 76.5
Subtract diff. bet. nos. oct. in E + 2 (8) and

E + 3 (9; = 28 + 8 = . . . . . 36

Remainder 40.5-
Remainder being positive and less than next

difference, subtract — - 1.5

b 6

Remainder 39

Multiply remainder by .... 6

Product = remainder over, required . . 234
S'ubtract same from given number . . . 963

Remainder =. greatest third power required . 729
n = 9.

With the same given number, let the greater of the
two numbers in the table, which are equally near the
quotient, be taken as the nearest number.

Given number 963

Divide it by 6. Quotient 160.5

Subtract nearest no. (E . 12) . . . 286

First remainder - 125.5
Subtract diff. bet. nos. oct. in E (12) and

E - 1 (11), as in table . . - 66

Remainder — 59.5
Subtract diff. bet. nos. oct. in E — 1 (11) and

E - 2 (10) = - 66 - (- 11) = - . . - 55

Remainder — 4.5

QUESTIONS IN MATHEMATICS. 77

Remainder still being negative, subtract diff.
bet. nos. oct. in E - 2 (10) and E - 3 (9)
= — 65 — (— 10) = . . . . - 45

Remainder 40.5

Remainder being now positive, and greater
than the quotient of the least edge which
has come into the process, E — 3, divided

by 6, subtract -'— — = - - = . . . 1.5

Remainder 39

Multiply remainder by 6

Product = remainder over, required . . 234
Subtract same from given number . . 963

Remainder = greatest third power required 729
n = 9.

Let the given number be

Divide it by 6. Quotient 286

Subtract nearest no. oct. in table (E . 12) . 286

First remainder 0
Subtract din3, bet. nos. oct. in E (12) and

E - 1 (11), as in table . . . . - 66

Remainder 66
Remainder being positive and greater than

E - 1 11 ,11

— - — = — , subtract — = . . 1.833

bo b

Remainder 64.166

Multiply remainder by .... 6

Given number = (E — I)3 = II3 + rem. over 385

78 QUESTIONS IN MATHEMATICS.

It is also equal to E3 — E = 123 -- 12, being the point
at which the first remainder becomes 0.

Again let the given number be . . . 1729

Divide it by 6. Quotient 288.166

Subtract nearest no. oct. in table (E . 12) . 286

First remainder 2.166
Remainder being a positive quantity but
not exceeding the number shown in the
second column of differences in the table,

E 12
subtract -5- = -5- = • • • • 2

b o

Remainder .166

Multiply remainder by .... 6

Given number = E3 = 12s + remainder over 1.

E3 being the point at which subtraction from the first

-p
remainder begins with a positive quantity, namely, -^ .

Again, let the given number be . . . 1727

Divide it by 6. Quotient 287.833

Subtract nearest no. oct. in table (E . 12) . 286

First remainder 1.833
Remainder being positive but less than

E 12

r = — , subtract diff. bet. nos. oct. in
6 6

E (12) and E - 1 (11), as in table . . - 66

Remainder 67.833

Subtract ?L* = ^ = 1.833

Q O

Remainder 66

QFESTIONS IX MATHEMATICS. 79

Multiply remainder by .... 6
Given number = (E — I)3 = II3 + rem. over 396

It is also equal to E3 — 1 = 123 -- 1, being the point
at which subtraction from the iirst remainder begins with
a negative quantity.

Let it be required to find the edge of a regular tetra
hedron of which the volume is 140608

Divide it by 6. Quotient 23434.666

Subtract nearest no. oct. in table (E . 54) 26235

First remainder - 2800.333
Subtract diff. bet. nos. oct. in 54 and 53,

as in table . — 1431

Remainder - 1369.333
Subtract diff. bet. nos. oct. in 53 and 52

- 1431 - (- 53) = . . . - 1378

Remainder 8.666
Remainder being now positive and equal

to 5?, subtract — = , 8.666
6 6

n = 52.

Let it be required to find the greatest third power

and remainder over, if any, contained in the given num
ber 157463, and also the root of such power.

Given number 157463

Divide it by 6. Quotient 26243.833

Subtract nearest no. oct. in table (E . 54) 26235

First remainder 8.833

QUESTIONS IN MATHEMATICS.

Remainder, although positive, not being

54
equal to — , subtract diff. bet. nos.

oct. in 54 and 53, as in table . . - 1431

Remainder 1439.833

Subtract -r = .... 8.833
6

1431
6

Remainder
Multiply remainder by .

Product = remainder over, required
Subtract product from given number

Rem. = greatest third power required . 148877
•n = 53.

8586
157463

Again, let the given number be .

Divide it by 6. Quotient

Subtract nearest no. oct. (E , 600)

First remainder

Subtract diff. bet. nos. oct. in 600
and 601 . . . . .

Remainder

, 601
Subtract -- =

218000000

36333333.333

35999900

333433.333

180300

153133.333
100.166

153033.166
6

Remainder
Multiply remainder by ...

Product = remainder over, required .
Subtract product from given number

Rem. = greatest third power required 217081801
n — 601.

918199
218000000

QUESTIONS IN MATHEMATICS. 81

Thus, with the aid of the table, every exact third root
consisting of one, two, three, or four figures, and the
first six of five figures, can be readily found. Beyond
10005 it is manifest could also be found by continued
subtractions of differences between the numbers of octa-
hedra in E + 3 and E + 4, and between those in E + 4
and E + 5, and so on, but this procedure would soon
become very irksome.

To construct a table by means of which every exact
third root of five figures and the first few of six could be
found, would require that it should be prolonged up to
16667 lines.

But the fifth figure may be reached approximately by
means of the table, and afterward exactly by calculation,
as follows :

Let it be required to find the third root of the given
number 388441751777344

Divide it by 6. Quotient 64740291962890.666

The first four figures of the root will now be found by

mere inspection of the table to be 7296. The number

of octahedra in a regular tetrahedron of the edge of

72960 may be found therefrom as follows :

No. oct. in 7296 . . . 64729648840

Add 5 = 1216

and affix 3 ciphers to the sum . 64729645056000

This is the quotient of S di
vided by 6 when n = 72960.

Subtract ~ = , 121 6C

6 :

No. oct. in 72960 64729645043840

82 QUESTIONS IN MATHEMATICS.

Comparing now the difference by which the quotient
of the given number divided by 6 exceeds the number
of octahedra thus found with the difference between
the numbers of octahedra in tetrahedra of edges 72960
and 72961, and observing that the product of the latter
multiplied by 4 is nearly equal to the former, it will
be evident that if the given number is a perfect third
power, the fifth figure of its root will be 4, and the pro
cess to demonstrate it may be further continued as fol
lows :

To the number of octahedra . 64729645043840
found as above, add as follows :

Diff. bet. nos. oct. in 72960 and
72961 (= 72960 x 72961 =

2661617280) x 4 = . . 10646469120

+ 72961 x 3 = . . 218883

+ 72962 x 2 = . . 145924

+ 72963 x 1 = 72963

Subtract sum . . • 64740291950730

from quotient of given num
ber divided by 6, as above . 64740291962890:666

Remainder 12160.666

79QB4

Subtract - - = . . . 12160.666
6

n = 72964.

From the number of octahedra in 72960 and differ
ences between that and the numbers in 72959 and 72961

QUESTIONS IN MATHEMATICS. 83

may he found the exact third roots of all perfect third
powers from 729303 up to and including 729903 and the
third roots and last remainders of all intermediate num
bers. Beginning with 729303 and going back to and in
cluding 728713, the first four figures of the root would be
found by means of the 1215th line of the table, and be
ginning with 729913 and going forward up to and includ
ing 730503, they would be found by means of the 1217th
line. The range is thus limited to 30 i*n each direction.

But the range, in so far as the calculation required is
concerned, is in fact limited to 10 in one direction only.
Thus, if the given number in the foregoing illustration
had been such that by inspection of the table and com
parison of the quotient with the number of octahedra in
7296, in connection with the differences given in the same
line, it had been seen that the first four figures were
probably 7295 or 7297, then the number in 72950 or 72970
would have been sought instead of that in 72960, and in
like manner in respect to 7294 or 7298, &c.

If the given number in the foregoing example had
been such that the fifth figure of the root would have
been 9, then there would have been 9 multiplications
towards the close of the process instead of 4 as in the
example, The process in respect to 8 of these multipli
cations (3 in the example), namely, all except the first,
can be shortened and will be found to apply to finding
all the required figures of the root. Such shortening of
the process in the case under consideration consisted in
finding the product of 72960 by the sum of the multi
pliers 8, 7, 6, 5, 4, 3, 2, 1 (3, 2, I in the foregoing exam
ple) and adding thereto the term of the following series,

84 QUESTIONS IN MATHEMATICS.

the number of which is equal to the number of the re
quired figure of the root, instead of finding successive
products by such multipliers as in the example, such
product by the sum of the multipliers with the term of
the series added thereto being equal to the sum of such
successive products. The sum of the multipliers is equal
to the quotient by 2 of the product of the highest of such
multipliers by itself + 1.

No*, of terms? 12345678 9, &c.

Series, 0 1 4 10 20 35 56 84 120, &c.
1st or. diff's 136 10 15 21 28 36, &c.

sdor. diff's. 234 5 6 7 8, &c.

The series was found as follows : The author consid
ered the required fifth figure of the root to be successively

2, 3, 4, 5, 6, 7, 8 and 9, and found in each case (except
the first) the sum of the products of the multiplications
as in the foregoing example (that is, in each case, of
multiplications equal in number to the number of the
required figure of the root less 1), namely, in the case of

3, of 72961 x 2 and of 72962 x 1, in the case of 4, as in
the foregoing example, and so on up to the case of 9,
in which the multiplications began with 72961 x 8 and
ended with 72968 x 1. Then dividing 72961 in the case
of 2 and the sum of the products in each case there
after by 72961 the quotients were found to be the sums
respectively of the multipliers with remainders consti
tuting the series as above given. The term of the series
in each case was then found to be the difference between
the sum of the products and the product of 72960 by the
sum of the multipliers.

QUESTIONS IX MATHEMATICS. 85

The term of the series required in the process as to
any given number of which the third root is sought may
be found by means of the term of the first order of dif
ferences on the left hand of the required term of the
series, and of the term of the second order of differences
under such required term, and of the number of such
required term.

Let T signify the required term of the series, t the
number of the term, a the term of the first order of dif
ferences as above, and 5 the term of the second order.

Then

b =t

V b
= 2"" 2"

But there is a shorter method. The foregoing series
is the series of octahedra contained in regular tetrahedra
of edges beginning with I and continued according to the
series of natural numbers. (See Analysis on page 68.)
The required term of the series is therefore equal to the
number of octahedra contained in a regular tetrahedron,
the edge of which is equal to the number of the term
(which, as before stated, is equal to the number of the
required figure or figures of the root), and may be found
as shown on page 69. Thus,

'Y — f. v _

< 6 " (T

Let it be required to find the third root of the given
number ...... 388521613429209

Quotient by 6 64753602238201.5

QUESTIONS IN MATHEMATICS.

The first four figures of the root will now be found
by inspection of the table to be 7296, as in the next pre
ceding example (on pages 81 and 82), and the number of
octahedra in 72960 may be found as shown in that ex
ample, and by comparison of differences also as in that
example, the probability that 9 will be the fifth figure
of the root will appear. The process will now be further
conducted as follows :

Number of octahedra in a regular tetrahedron of edge

72960 as found on page 81 .

+ 2661617280 (see page 82) x 9 =

8x9

64729645043840
23954555520

2626560

+ 9th term of series

120

Subtract sum 64753602226040

from quotient by 6 of given number . 64753602238201.5

Remainder

72969

Subtract

12161.5
12161.5

n = 72969.

Let the given number of which the third root is re
quired be . . . 1000660148211968148200660001

Quotient by 6 . 166776691368661358033443333.5

The 'first five figures of the root will be found by in
spection of the table to be 10002.

QUESTIONS IN MATHEMATICS. 87

No. of oct, in 10002 166766685001
Add - = . . 1667

and affix 15 ciphers . . 166766686668000000000000000

This is the quotient
of S divided by 6 when
n = 1000200000.

Subtract ^ = 166700000

No. of oct. in

1000200000 . . 166766686667999999833300000

Diff. bet. nos. oct. in. 1000200000 and 1000200001 =
100^0^0^K)002(^OW = 500200()20500100000.

Comparing differences as before, the number of the five
required figures of the root will be found to be 20001.

No. oct, in 1000200000 . 1667666.86667999999833300000
+ diff. as above x 20001 = 10004500610022500100000
+ 1000200000 x
2000*0 x 20001

200050002000000000

200012 x - = 1333533340000
6

2
20001st term of series =

. 166776691368661357866740000

from quo. of giv. no. by 6 1 66776691368661358033443333.5

Remainder 166703333.5

Subtract 1000220001 = 166703333.5

6 —

n = 1000220001.

QUESTIONS IN MATHEMATICS.

It will now be manifest upon due consideration of the
foregoing examples, that what can be done by means of
the first five or four figures of the root found by the table
could as well have been done by means of the first three
or two figures found by shorter tables, or by means of
only the first figure found directly from the given num
ber and without the aid of any table.

Let it be required to find the third root of the given
number 606569944625.

By the usual method of pointing off the figures of the
given number it will be seen that the first figure of the
root is 8.

The number of octahedra in a regular tetrahedron of

edge 8 = 82 x ~ - ~ = 84.
6 6

No. oct. in 8 . .

and affix 9 3's . . .
removing the decimal point 9 places to
the right.

This is the quotient of S divided
by 6, when n — 8000.

fli

Subtract — =
No. oct. in 8000

1.333
85333333333.333

1333.333

85333332000

TV-ff U • QAAA A CAA1 800° X 8001

Diff. bet. nos. oct. in 8000 and 8001 = — - —

= 32004000.

By comparing differences as before the number of the
three required figures of the root will be found to be

QUESTIONS IN MATHEMATICS. 89

No. oct. in 8000 .... 85333332000
+ diff. as above x 465 = . . 14881860000

Af\A v

+ 8000 x = . . . 863040000

+ 465th term of series =

4652 x ~ - ~ = 16757360

6 0 -

Subtract sum ..... 101094989360
from quo. of given no. by 6 . . . 101094990770.833

Remainder 1410.833

84fi5

Subtract - - = . . . 1410.833

n = 8465.

But there is still another and a much simpler process.

To the first figure of the root, as found from the
given number, affix as many ciphers as with the first
figure will make up the number of places of figures in
the required root, as indicated by the pointing off of the
figures of the given number, and assuming the number
thus formed to be the number of the edge of a regular
tetrahedron, find the number of octahedra contained in
such regular tetrahedron. Then from the quotient of
the given number divided by 6 subtract the number of
octahedra thus found, and set down the remainder. Find
then the difference between the number of octahedra so
found and the number contained in a regular tetrahedron
of the next succeeding edge, and divide the remainder
found as above by such difference.

The quotient as a whole number (disregarding frac
tions) resulting from such division, if it shall consist of
places of figures equal in number to the number of places-

'90 QUESTIONS IN MATHEMATICS.

of the required figures of the root, will be such required
figures; but if it be less, then prefix thereto one or
more ciphers, as may be necessary to make up such num
ber of places, and such cipher or ciphers with the quo
tient will be such required figures.

Find now the number of octahedra contained in a
regular tetrahedron of edge equal to the whole root as
thus found, and multiply such number of octahedra by 6,
and to the product add the number of the root.

The sum will be equal to the given number if the
latter be a perfect third power ; but if such sum be
less, then subtract it from the given number and the
remainder will be the excess of the given number over
and above the greatest third power therein contained.

To illustrate. — Let it be required to find the third root
of the greatest third power and remainder over, if any,
contained in the given number 1006012009.

By pointing off the figures of the given number, the
first figure of the root is found to be 1, and the number
of places of required figures 3.

The number of octahedra contained in a regular

tetrahedron of edge 1000 is equal to 10002 x -

166666500

Subtract same from quotient of given
no. by 6 . . . . . . . 167668668.166

Remainder 1002168.166

1000 x 1001
Diff. bet. nos. oct. in 1000 and 1001 = -

= 500500.

QUESTIONS IN MATHEMATICS. 91

1002168.166 =

500500

Prefixing two ciphers to this quotient, the required
root is found to be 1002.

No. oct. in 1002 - 10022 x ~ - ~ = 167668501

b o

x 6

= 1006011006

+ number of root 1002

Subtract sum . . . 1006012008
from given number 1006012009

Given no. = 10023 + remainder . . 1

The table having been thus found to be entirely un
necessary for the purpose of evolution to the third degree,
it will probably be asked, Why was it introduced at all,
and why the long description and illustrations following
it ? The answer is, that the shortened processes were not
found by the author until after the whole book was in
type and nearly ready for the press, awaiting only some
final corrections of a few of the plates. The author then
found himself in a quandary as to what course he should
pursue, whether to strike out the table and, all following
it down to the end of the second paragraph on page 83
(which would have involved the necessity of striking out
also all the following pages down to § 17, on page 107)
and write a description of the process as finally reached,
and there leave the subject, or to open the book and
take in the description of the shortened processes to their
ultimate result (striking out, however, a cumbrous method
of finding the fifth and subsequent figures of the root

92 QUESTIONS IN MATHEMATICS.

by means of farther tables, one for each figure), and
accompany it with this explanation.

He adopted the latter alternative for the reason that
the book as it now stands shows the entire course of in
vestigation by which he reached the final result, and with
out which, perhaps, such result would never have been
attained, and for the further reason that, as with the aid
of the table the operations of involution to the third de
gree and of both evolution and involution to the second
degree may be performed (as hereinafter shown) with
much greater facility than by the usual arithmetical
processes, so it may be, that the table may be found of
service in respect to other operations of which the author
has no knowledge. ,

The third power of any number up to 10002, and a
few numbers beyond, may be found with the aid of the
table by a shorter process than by taking the number
three times as a factor.

Thus, let it be required to find the third power, or
volume of a regular tetrahedron of the edge of 68.
The nearest edge in the table is 66.
No. oct. in 66, as in table . . . . 4790,5
+ diff. bet. 66 and 67, as in 2d col. diffs . 2211
+ diff. bet. 67 and 68 (= 2211 + 67) . . '2278

= . . . . . . '. . . 52394

x . . . ' 6-

= . . . . ' . . ... . 314364

4- given number . . ... . 68

= . . . . . . 314432

QUESTIONS IN MATHEMATICS.

To find the third power of 4757.
Nearest edge in table 4758.

No. oct. in 4758

+ diff. bet. 4758 and 4757, as in 1st col.
diflTs

17952380459

- 11316903

17941063556
6

•f given number

== 47573.

107646381336

4757

107646386093

If the given number should be higher than a few
beyond 10002, or be of six or more figures, a process
could undoubtedly be devised to find the power. The
author has not attempted to find such process.

By means of the foregoing table may also be found
the second root of all perfect second powers and the root
of the greatest second power and remainder over, con
tained in all intermediate numbers when such root con
sists of one, two, three, or four figures, or is one of the
first six of five figures.

The sum of the differences between the number of
octahedra in a regular tetrahedron of any given edge and
those of the next preceding and succeeding edges respect
ively, considered both as positive quantities, is equal to
the second power of the number of the given edge.

1st diff. 2d diff.

n (n — 1) , n (n 4- 1)

v / I \ ' — - /Yi«

•• iv •

94 QUESTIONS IN MATHEMATICS.

To find the second root of any given perfect second
power, or such root of the greatest second power and re
mainder over, contained in any given number, divide
the given number by 2, and look in the first column of
differences in the table for the number nearest the quo
tient. If there are two numbers in the first column of
differences equally near the quotient, either may be taken.
This can occur when the given number is an exact second
power only once, namely, in the case of 92.

Note the edge (root) in the same line with the nearest
number taken.

Compare the nearest number found in the first col
umn of differences with the quotient, and if it be less,
observe whether the excess of the quotient is equal to

T^ S T^

~ and is not greater than - — .

& li

If such excess is within these limits (both inclusive),
then the numbers given in the two columns of differences
in the table are the differences sought, the sum of which
is equal to the given perfect second power, or to the
greatest second power contained in the given number,
and E is the required root.

If such excess is less than — , or if the nearest num-

ber exceeds the quotient, then set down the nearest num
ber so found and subtract from it E — 1. If the re
mainder exceeds the quotient, subtract therefrom E — 2,
and if, as thus diminished, it still exceeds the quotient,
subtract from it E — 3. The quotient will now exceed
the remainder, and the latter will be the first difference
sought, and the next preceding remainder, or if there

QUESTIONS IX MATHEMATICS. 95

shall have been but one subtraction, then the number
taken from the table will be the second difference sought,
and the last number subtracted (E — 3 or 2 or 1, as the
case may be) will be the required root.

If the excess of the quotient over the nearest number

3 V
is greater than - — , then go to the second column of dif-

ferences, same line, and set down the number therein
found (which is the second difference in E and first in
E + 1) and add thereto E + 1. If the sitm be less than
the quotient add to it E + 2, and if as thus increased it
be still less, add to it E + 3. The sum will now exceed
the quotient and will be the second difference sought,
and the next preceding sum, or if there shall have been
but one addition, then the number taken from the table
will be the first difference sought, and the last number
added (E + 3 or 2 or 1, as the case may be) will be the
required root.

Find in either case the sum of the two differences
and subtract it from the given number. If there be no
remainder, the given number ^s a perfect second power ;
but if there be, then such remainder is the excess of the

given number over the greatest second power therein
contained, and will be the last remainder that would be
found in the usual arithmetical process of extracting
the second root.

To illustrate. Let the given number of which the
second root is required be ..'... 36

Divide it by 2. Quotient, 18.

Nearest no. in 1st. col. diff's in table is 15
(E . 6). Excess of quotient over same being

•96 QUESTIONS IX MATHEMATICS.

E
•equal to -^, the two differences in the table are

the differences sought.

1st diff. in E (6) . . . . .15

2d diff. in E (6) . . . . . .2^

Find sum of diffs ...... 36

and subtract same from given number

There being no remainder, ....

the given number is an exact second

power, of which
E = 6 is the required root.

Again, let the given number be ... 48

Divide it by 2. Quotient, 24.

Nearest no. in 1st col. difFs in table is 15
(E . 6). Excess of quotient over same not ex
ceeding — , the two differences in the table
ic

are the differences sought.

1st diff. in E (6) . . . . . .15

2d diff. in E (6) ... . . . . 21_

Find sum of diff's 36

and subtract same from given number.
Given number = E' = 62 -f remainder . 12

Again, let the given number be ... 81

Divide it by 2. Quotient, 40.5.

Two numbers in the first column of differ
ences are eqimlly near the quotient, viz., 15
(E . 6) and 66 (E . 12). Let the latter be first
taken as the nearest number.

QUESTIONS IN MATHEMATICS. 97

Nearest number (E .12) . . \ . 66
Same exceeding quotient, subtract E — 1 — 11

Remainder = 1st diff. in E - 1 (11) . . 55
Same exceeding quotient, subtract E — 2 = 10

Remainder = 1st diff. in E - 2 (10) . . 45
Same still exceeding quotient, sub. E — 3 = 9

Remainder = 1st diff. in E - 3 (9) . . 36

Remainder being now less than quotient,
bring down next preceding remainder =
2d diff. in E - 3 (9) . . . ' . .45

Find sum of diff's ...... 81

and subtract same from given number .

E — 3 = 9 is the required root.

With the same given number, let the first of the two
numbers which are equally near the quotient be now
taken as the nearest number.

Given number ...... 81

Divide it by 2. Quotient, 40.5.

Nearest no, (E .6) 15. Excess of quotient

over same being greater than - — , take

no. in 2d col. diff's . . . .21

+ l= ..... .7

Sum = 2d diff. in E + 1 (7) . . . .28
Same being less than quotient, add E + 2 = 8

Sum = 2d diff. in E + 2 (8) . ." . .36
Same being still less than quotient, add
E + 3 = . . . .9

Sum = 2d diff. in E + 3 (9) . . . .45

QUESTIONS IN MATHEMATICS.

Sum being now greater than quotient, bring
down next preceding sum = 1st diff. in

E + 3(9) . -'. ...

Find sum of diff's ....'.

and subtract same from given number .
E + 3 = 9 is the required root.

Again, let the given number be ...
Quotient by 2. 11324040.
Nearest no. in 1st. col. diff's (E . 4758)
11316903. Excess of quotient over this

number not exceeding - — , the diff's in

table are those sought.
1st diff. in 4758 . . . . 11316903
2d diff. in 4758 . ... . 11321661

Find their sum ....

and subtract same from given no.
Given no. = 47582 + remainder .

Again let the given number be .
Quotient by 2. 11324041.
Nearest no. in 1st col. diff's (E . 4758)
11316903. Excess of quotient over this

S F

number being greater than -=-, take

no. in 2d col. . . . 11321661

Add E + 1 = . . . . 4759

Sum = 2d diff. in 4759 . . 11326420
Bring down no. taken from table

= 1st diff. in 4759 . 11321661

22648080

22638564

9516

22648082

QUESTIONS IN MATHEMATICS. 99

Find sum of (Jiffs . . . . . 22648081
and subtract same from given no.

Given no. = 4759* + remainder ... 1

The fifth and subsequent figures of the second root
could probably be found by a process analogous to
that for finding the fifth and subsequent figures of the
third root. The author has made no attempt to find
such process.

The second power of any number given in the last
column of the table is the sum of the differences in
the same line considered both as positive quantities.
The second power of any number intermediate between
any two consecutive numbers in the last column of
the table may be found with the aid of the table, as
follows :

Let it be required to find the second power of 598 in
termediate between 594 and 600, the nearest number
being 600.

No. in 1st col. diff' sin 600 . . . . 179700

599

= (2d diff. in 598) 179101

+ (179101

- 598 = 1st diff. in 598 =) . 178503

= . . . . '.'•• .... 357604
= 5982.

Again, let the given number be 7299 intermediate be
tween and equally near 7296 and 7302 in the last column
of the table. Take 7296.

100 QUESTIONS IN MATHEMATICS^

No. in 2d col. diff s in 7296 . . . 26619456

+ . . . . . . . . . 7297

+ 7298

= (1st diff. in 7299) 26634051

+ (26634051 + 7299 = 2d diff. in 7299 =) . 26641350

. . . ' '. . . . . 53275401

= 72992.

In the case of higher numbers than those to which
the table is applicable, resort would of course be neces
sary to a further process which could probably be de
vised for the purpose. As before, the author has made
no attempt to find such process.

It may perhaps be said that the operations of evolution
and involution are performed by means of logarithms
which apply as well to finding higher powers and their
roots, and that the foregoing processes apply only up to
the third degree, and that as in respect to such higher
powers there can be no physical representation, so there
can be no tables constructed nor processes devised anal
ogous to the foregoing table and processes for the pur
pose of finding them or their roots. But the foregoing
table and processes are not given as a proposed substi
tute for the table of, and processes by, logarithms, in so
far as they apply, but as reasons and illustrations in
support of an affirmative answer to the main question
propounded in this book. They could never have been
discovered by an analysis of the cube.

In the description of the method of finding the third
root by means of the table, it is stated that it can never

QUESTIONS IN MATHEMATICS. 101

occur when the given number is an exact third power,
that there will be two numbers of octahedra in the table
equally near the quotient of the given number divided
by 6.

. The series of numbers in respect to such quotient of
each of which there are in the table two numbers of octa
hedra equally near, may be said to begin with 105, the
quotient of which divided by 6 is 17.5, to which 0 and
35 (the first number of octahedra in the table) are equally
near, and in respect to this number there is this pecul
iarity, which will not again occur in the forward direc
tion of the series. The greatest exact third power con
tained in 105 is 64 = 43, and the remainder over is 41.
But 105 is also equal to 3s + 78, and must be considered
as such with reference to the series, which is as follows :

105 33 + 78

963 = 93 + 234

3765 = 153 + 390

9807 = 213 + 546

20385 = 273 + 702

* * *

806775 = 933 + 2418

972873 = 993 + 2574

1160355 — 105s + 2730

1370517 = 1113 + 2886

and so on.

The term of this series lying between any two con
secutive numbers of octahedra in the table may be found
by adding together such numbers and multiplying their
sum by 3.

Thus, (35 + 286) x 3 = 963.

102 QUESTIONS IN MATHEMATICS.

All the terms of the third order of differences of the
series are equal, each being 1296 = 64.

The difference between the roots of each two consecu
tive greatest third powers (the first being considered
such as above stated) is in all cases 6, and that between
each two consecutive remainders over, is in all cases 156.
The remainder over, in the first term of the series, it
will be observed, is one-half such difference.

This difference, 156, is made up of equal differences
between each two consecutive terms of an expanded series
intermediate between any two consecutive terms of the
above series, as will be seen by the following analysis of
the expanded series intermediate between the first and
second terms, to be considered in connection with the
first two lines of the table expanded, as subsequently
shown.

105

= 33

+ 78

168

= 43

+ 104

255

= 53

+ 130

372

= 63

+ 156

525

__ »s

rf- 182

720

= 83

+ 208

963

= 93

+ 234

All the terms of the third order of differences of the
expanded series are equal, each being 6, and the differ
ence between any two consecutive remainders over, in
the last column is in each case 26.

The following table exhibits the first two lines of the
table of octahedra fully expanded, the last column show
ing the edges from 1 up to 12 instead of in multiples of 6,
as in the original table, but with the columns of differ-

QUESTIONS IN MATHEMATICS.

103

ences omitted to give place to a diagram showing two
numbers of octahedra, six lines apart, equally near the
quotient of each term of the foregoing expanded series
divided by 6.

TABLE OF OCTAHEDRA.

(Expanded.}

VALUE OP

EDGE

A continuation of the expanded series backward will
show that its first term is 27 and the first remainder
over, 26. Thus,

27 = I3

60 = 23

105 = 33

26
52

104 QUESTIONS IN MATHEMATICS.

The number 26 has been hereinbefore (page 53) shown
to be the base of the differences between the volumes of
figures of any two consecutive orders in the table of
natural involution.

It is also the apparent volume of a tetrahedron of
edge 3 as calculated from an external view as hereinbe
fore described (page 66), the concealed tetrahedron being
the central figure, and is the difference between the vol
ume of the tetrahedron of the first order and that of one
of the second order in the suggested table based upon
the tetrahedron as the central figure (page 56) and the
base of all subsequent differences between the volumes in
any two orders in the same table.

In the description of the method of finding the second
root by means of the table, it is stated that it can occur
but once when the given number is an exact second
power, namely, in the case of 92, that there will be two
numbers in the first column of differences in the table
equally near the quotient of the given number divided
by 2.

The series of numbers in respect to the quotient of
each of which divided by 2, there are two numbers in
the first column of differences in the table equally near,
is as follows :

15 = 32 + 6

81 = 92

219 = lo2 6.

429 = 212 12

711 = 27* 18

and so on.

QUESTIONS UN' MATHEMATICS. 105

The term of this series lying between any two con
secutive numbers in the first column of differences in the
table is the sum of such numbers.

Thus, 15 + 66 = 81.

All the terms of the second order of differences of the
above series are equal, each being 72, and the difference
between any two remainders over in the last column is
in each case 6.

The expanded series intermediate between the first
two terms of the foregoing series and continued beyond,
is as follows :

15 = 32 + 6

21 = 42 + 5

29 = 52 + 4

39 = 62 + 3

51 = 72 + 2

G5 = 82 + I1

81 = j)2

99 = 102 1

119 II2 2

141 = 12* 3

and so on.

All the terms of the second order of differences of the
expanded series are equal, each being 2, and the differ
ence between any two remainders over, in the last column
is in each case 1.

The author notes the following further observations
in the analysis of the several courses of a regular tetra
hedron and of a series of whole figures, as shown on
page 68.

100 QUESTIONS IN MATHEMATICS.

The figures in the units' places in the first five, ten,
or twenty lines of the table (considered as continued)
are as follows :

In respect to Courses.

Of tetrahedra 1.3.7.3.'!.

Of octahedra 0. 1. 3. 6. 0. 5. 1. 8. 0. 5. 5. G. 8. 1.5.0.0.3.1.0.

Of vols. of oct 0.4.2.4.0.

Of total vols 1.7.9.7.1.

In respect to Whole Figures.

Of tetrahedra 1.4.1.4.5.6.9.6,9.0.

Of octahedra 0.1.4.0.0.5.6.4.0.5.0.0.4.5.0.0.6.9.0.0.

Of vols. of oct 0.4.6.0.0.

Of total vols 1.8.7.4.5.6.3.2.9.0.

and they are the same in each succeeding five, ten, or
twenty lines, as the case may be. The sequence of the
figures in respect to the courses read backward is the
same as when read forward.

The numbers of the tetrahedra and octahedra appear
ing on the outer faces of the several courses (see pages
65 and 66) constitute two like series, but beginning with
course 1 as to the tetrahedra and with course 2 as to the
octahedra. The terms of these series in any course are

equal to n (^-\-.5j as to the tetrahedra and to n (- — .5)

as to the octahedra, and such terms are the differences
between the numbers of octahedra contained in the whole
figure of like edge as the course and in whole figures of
the next succeeding and preceding edges.

In the second order of differences of the two series of
numbers of tetrahedra and octahedra respectively con
tained in the several courses, and in the third order of

QUESTIONS IN MATHEMATICS. 107

differences of the two like series contained in whole
figures, the terms become equal, and are in each case 2 in
respect to the series of tetrahedra and 1 in respect to
those of octahedra.

The terms of the third order of differences of the
series of numbers of octahedra in the table on page 71
become equal, and are in each case 216 = 63.

§ 17. If the octahedron be revolved one-fourth of a
revolution on each of its three axes, its moving points
will describe three circles, each bisecting the other two.
If such circles are delineated on the surface of the cir
cumscribed sphere of the octahedron, they will be great
circles of the sphere, and will divide such surface into
eight equal trirectangular spherical triangles, represent
ing the primary division of the surface of the sphere as
delineated by geometers, geographers, and astronomers.

The further division of the surface of the sphere is
made by geographers and astronomers by great circles
representing meridional lines by which longitude is reck
oned, and small circles parallel to the equator by which
latitude is reckoned. These artificial circles with the
three primary natural ones by their intersections divide
the surface of the sphere into quadrilaterals (varying in
number according to the number of the artificial circles),
except immediately about the poles, where the division
is into isosceles triangles.

These quadrilaterals are all irregular, and increase in
the dimensions of two of their sides as they approach the
equator from either pole, and the degrees of longitude as
they are measured on different parallels of latitude on

108 QUESTIONS IN MATHEMATICS.

the surface of the earth, considered as the sphere, vary
in length from sixty geographical miles on the equator
to zero at the poles.

Let the circumscribed sphere of the octahedron,
having delineated thereon only the three primary great
circles, be considered as having the points of intersec
tion of such circles (which are the points of the inscribed
octahedron) designated by the logical symbols succes
sively in the three diiferent ways shown in the three
diagrams on pages 40, 41, and 42.

In each case the two complete processes of reasoning
(complete in the logical sense but not in their combina
tions on the faces of the octahedron, or the surface of
its circumscribed sphere), have their terms of beginning
(of the progressive sorites in each case) at the poles re
spectively, but the ultimate terms (of both the progress
ive and regressive sorites) in respect to each process will
be found as follows :

In each of the first and second cases, that of the pro
gressive sorites at one point and that of the regressive at
another, both on the equatorial line, the designations, in
the first case, being on the opposite side of the line from
the point of beginning, but, in the second case, on the
same side as the point of beginning.

In the third case, that of the progressive sorites at a
point on the equatorial line, with the designation on the
same side as the point of beginning, and that of the
regressive sorites at the opposite pole from the point of
beginning.

In the first case, no sorites has its two syllogisms on
adjacent faces ; in the second, one sorites of each process

QUESTIONS IN MATHEMATICS. 109

only, and in the third both ; but in the third case, the
two sorites of each of the processes are not combined in
regular order respectively, and procedure cannot be made
from one to the other, and the two processes are not, as it
would seem they should be, on hemispheres bounded on
the surface by the equatorial line, but are on hemispheres
bounded by a great circle passing through the poles. ,

§ 18. If the oct'astron be revolved one-third of a revo
lution on each of its four axes, its moving points will
describe eight circles. If such circles are delineated upon
the surface of the circumscribed sphere of the oct'astron,
each two parallel ones described by each partial revolu
tion will divide such surface into three zones, and the
planes of each such two parallel circles will trisect the
related axis of the sphere. The eight circles, by their
mutual intersections, divide the surface of the sphere
into forty-two figures, of which eighteen are quadri
laterals, and twenty-four isosceles triangles. Twelve of
the eighteen quadrilaterals are rhombs and six, squares.

All the figures of the same kind are equal to each other.

Let now such a sphere with circles so delineated be
considered as in hand, and let those points on the sur
face, which are the points of the included oct'astron, be
marked with the symbols of the two complete processes of
reasoning, as the points of the oct'astron are marked as
shown in figures 17 and 18, on page 34, and in the descrip
tion following those figures, and let the axis, the extremi
ties of which are marked XXX and N' N' N', be regarded
as vertical and as the axis of revolution of the sphere.

Each point will be found designated by but one sym-

110 QUESTIONS IN MATHEMATICS.

bol instead of by two or four, as in the case of the cir
cumscribed sphere of the octahedron, and the two com
plete processes of reasoning, beginning at the poles re
spectively, instead of being disjoined or imperfectly
conjoined as in such case, will be found, the two sorites
of each process perfectly combined, and the two complete
processes perfectly conjoined, overlapping each other and
having their ultimate terms, both progressive and re
gressive of each process, at the same point, but of the
two processes considered relatively to each other at
opposite points of a diameter of the sphere, but on the
lines of two different circles, the bases of the middle
zone ; the term of beginning of each process at the poles
respectively being related to the ultimate term on the
circle farthest removed from such pole.

There will be no equatorial line. Three great circles
may be drawn dividing each of the rhombs into two
equal regular triangles, and each of the squares into four
equal right-angled triangles. These great circles would
be the boundaries of imaginary planes passing through
the centre of the sphere, in each of which planes two
axes and four edges of the octahedron included in tke
included oct'astron would lie, but neither of the circles
would be equatorial relatively to the axis of revolution
of the sphere, or to either of the other axes of the in
cluded oct'astron.

These great circles would, however, probably never be
required. To lay them down and thereby draw a diagonal
through each of the rhombs would be equivalent to ex
pressing the unexpressed conclusion of the first, which
is the unexpressed premise of the second of the series of

QUESTIONS IN MATHEMATICS. Ill

two syllogisms into which a simple sorites may be ex
panded. A simple sorites is as manifestly conclusive on
its face as a simple syllogism.

Let the reader now consider that the sphere which he
has in hand is the circumscribed sphere of an oct'astron
of the edge of 1, and that it is held with its axis, X N',
vertical, in such position below the eye that tlte great
triangle X J D is in full view. The following figure
will then be presented :

Fig. 34

One-third only of the surface of the sphere is plainly
visible, viz. : the zone about the point X' and bounded
by that one nearest such point of the two circles de
scribed by the moving points of the oct'astron in the
partial revolution of the sphere about the axis X' N.

Revolving the sphere one-half of a revolution, the
following figure will present itself :

112

QUESTIONS IN MATHEMATICS.
Fig 35

If now the sphere be lifted up to a position as far
above as it was previously held below the eye, the great
triangle N' D' J' will come in full view, and upon turning
it about, one-half of a revolution (or one-sixth or five-
sixths), the figure presented will be similar (except as to
symbols) to the foregoing, but with the two fully shown
intersecting figures in the external form of lunes (and
which will be herein called lunes) on the middle and lower
instead of the middle and upper zones of the sphere.

By revolving the sphere, as held below the eye from
left to right, the great triangles X D N" and X J X will
successively come in view, and as held above the eye,
triangles IS"' J' X' and N' D' X' ; and by turning the fig
ure so that the vertical axis X N' shall be horizontal,
with the pole N' toward the eye, triangle N" D J will
come in view, and with the pole X toward the eye, tri
angle X' J' D'.

QUESTIONS 1^ MATHEMATICS. 113

The figures in the form of limes are directly over the
edges of the included oct'astron and separate the great
triangles of each process from each other. Each great
triangle is bounded by three lunes, and the outer line of
each lune (with reference to any great triangle), upon
being revolved on its chord, will coincide with the sur
face of the sphere until it shall reach and coincide with
the inner line, the side of the great triangle.

Each rhomb is common to two of the great triangles,
viz.: one of each of the two processes, descending and
ascending, and the points of their acute angles only are
designated, one by a symbol of one process and the
other by a symbol of the other, each of the extremes
of the process in either direction being connected by
three rhombs with the opposite extremes respectively .
and the two middle terms of the process in the other
direction.

The zone about either pole of any axis consists of
three rhombs, nine small triangles, and three squares,
and the middle zone consists of -six rhombs and six small
triangles. The area of each of the rhombs is therefore
equal to the sum of the areas of one of the small triangles
and one of the squares.

If the lines of but one of the complete processes of
reasoning, descending or ascending throughout, were
delineated upon the surface of the sphere, the figures
thereby produced would consist of four great triangles
and six lunes dividing the entire surface of the sphere.
Such lines would be delineated by the moving points
of a regular tetrahedron of the edge of 2, in its par
tial revolutions, as before described in respect to the

114 QUESTIONS IN MATHEMATICS.

oct'astron, the circumscribed sphere of such a tetra
hedron being equal to that of an oct'astron of the edge
of 1.

There would be but four circles and four points of
intersection of their lines, each point designated by one
of the symbols of the logical process. Each point would
be polar to one circle only, and there would be no oppo
site poles.

The following figure exhibits a zone of the sphere
with circles so delineated.

Fig. 36

The zone consists of a great triangle bounded by three
lunes, its area being one-third that of the surface of the
sphere. The great triangle with one-half of each of the
lunes by which it is bounded is equal in area to one-
fourth the surface of the sphere.

QUESTIONS IN MATHEMATICS. 115

§ 19. If partial sections be made into the body of the
sphere on which all the circles are delineated, follow
ing the planes of the circles down to the chords of the
arcs wrhich form the sides of the figures on the sur
face, and the spherical surfaces be cut off from each
figure, the resulting solid figure will be one of surpass
ing beauty and symmetry, the faces of which will be
the underlying planes of the figures on the surface of
the sphere.

The dimensions of the parts of each of the faces are
as follows : The sides of the rhombs are each .866, equal
to the altitude of the regular triangle (side 1) ; their acute
angles are 70° 31' 42", equal to the dihedral angle of a
regular tetrahedron, and their obtuse angles are 109°
28' 18", equal to the dihedral angle of a regular octahe
dron. If a diagonal were drawn bisecting the obtuse
angles, its length would be 1. The base of each of the
small triangles is .5, the other sides are each .866, the
angle opposite the base 33° 33' 30", and the other angles
each 73° 13' 15". The squares have their sides each .5,
and their angles, of course, each 90°.

§ 20. The polygons and angles on the surface of the
circumscribed sphere of the oct'astron are not spherical
polygons and angles as defined by geometers, which are
all bounded by arcs of great circles, and all have relation
to the centre of the sphere. But the angles of the great
triangles are the dihedral angles of the planes of the
circles by which they are formed, in like mariner as the
angles formed by the intersection of two great circles
are the dihedral angles of the planes of such circles, and

116 QUESTIONS IX MATHEMATICS.

the obtuse angles of the rhombs are also such dihedral
angles, and all such angles correspond to those of the
underlying planes of the rhombs.

In the case of the small triangles, the angles of the
underlying planes do not correspond to the dihedral
angles of the planes of the circles by which they are
formed. But by consideration of the preceding figures,
it will be manifest that the small triangles are not to be
considered by themselves, but each two with their inter
vening square as constituting a lune to be considered as
a whole in any process in which it may be involved, the
intersecting lines by which the lune is divided, being
part of a wholly distinct configuration, and not to be
taken into consideration in the process. The square is
thus entirely eliminated from the figures.

§ 21. If now the partial sections into the body of the
sphere be continued and made complete along the planes
of all the circles, the figure described in § 19 will be
divided into fifty-one parts corresponding to the number
of figures in an octahedron of the second order, viz. : the
nine perfect and regular figures of the included oct'as-
tron of the first order and parts, viz.: one-half of each of
the twelve octahedra first mentioned, one-fourth of each
of the twenty-four tetrahedra secondly mentioned, and
one-eighth of each of the six octahedra thirdly mentioned
in the description of the construction of the octahedron
of the second order hereinbefore contained (page 51).

The twelve octahedra first mentioned will have been
divided each by a plane passing from one point to the
opposite point through four faces adjacent in pairs and

QUESTIONS IN MATHEMATICS. 117

opposite in pairs, bisecting such faces and forming the
rhombs ; the twenty-four tetrahedia secondly mentioned
will have been divided each by a plane passing from one
of its points to the middle points of two sides of its op
posite face, and forming the small triangles, and the six
octahedra thirdly mentioned will have been divided each
by a plane passing through four of its faces having a
common point, beginning at the middle point of one edge
of the octahedron, and cutting such faces in lines parallel
to their sides opposite such common point, and forming
the squares.

It has been hereinbefore stated (on page 56) that the
tetrahedra in an octahedron of the second order are
analogues of compound logical processes through which
the two complete simple processes on the faces of the
oct'astron of the first order are brought into perfect
union. Such union consists in establishing the relation
between like extremes of the two processes. The sym
bols of such extremes as ultimately reached through such
compound processes do not, however, designate opposite
poles of two axes of the oct'astron considered as consist
ing of two intervolved tetrahedra, but are like symbols
of the extremes of the processes considered as conducted
on the faces of two of the superposed tetrahedra (as
described in § 11, on page 37), one in each direction,
and designate points of such tetrahedra which fall upon
the octahedron included in the oct'astron. Such points
do not come to the surface of the sphere, but the sym
bols designating them may be considered as brought to
such surface on the faces of the rhombs at their obtuse
angles, and by combinations of the ultimate results of

118 QUESTIONS IN MATHEMATICS.

the two processes considered as conducted upon the
tetrahedra as superposed and so brought to the surface
with the conclusions of the two processes considered
as conducted upon the tetrahedra as intervolved, the re
lation may be established between like extremes of the
two processes, the symbols of which designate opposite
poles of each of two axes of the oct'astron and of its
circumscribed sphere. All which will hereinafter (in
the appendix) be fully shown.

The circumscribed sphere of a tetrahedron or of an
oct'astron of the first order, and the inscribed sphere of
an octahedron of the second order, are equal to each
other, and the three considered as contained in an octa
hedron of the second order are identical, and may be
regarded as the emblem of triniunity.

The author concludes this, the main part of his trea
tise, with the following question :

Is not the delineation of the surface of the sphere,
produced by the revolution of the octahedron on its
three axes and supplemented by artificial lines, the bet
ter adapted for the description of the terrestrial sphere
for all the ordinary purposes of life ; and is not that
with natural lines only, produced by the revolution of
the oct'astron on its four axes, the better adapted for
all scientific purposes, and especially with reference to
the celestial sphere ?

APPENDIX.

THE following illustrations of the analogy between
compound logical processes and combinations of simple
geometrical figures could not have been introduced in
the foregoing treatise without breaking its continuity.
They are therefore given in the form of an appendix, but
to be considered as a part of the treatise.

The typical simple sorites of Concrete Logic is one
in which the magnus term is an individual thing which
can be predicated of nothing (except itself), the maximus
term, the highest term that can be predicated of the
magnus term, but of which nothing (except itself) can
be predicated, and the major-middle and minor-middle
terms, a genus and species respectively, of the first of
which the maximus term may be predicated, and the
second of which may be predicated of the magnus term,
and which are proximate to each other, so that the truth
of the proposition in which they are compared (the mid
dle premise) is readily recognized and admitted. (The
foregoing description is in the ascending direction, but,
by changing the expressions, it may be made applicable
also to the descending.)

In such a sorites there can be no additional terms in
troduced except those which are subsidiary, elucidating

120

APPENDIX.

either the relation between the major-middle and maxi-
mus terms, or that between the magnus and minor-
middle terms, or, by different new terms, both.

This will clearly appear (but by symbols indefinite in
material signification, of which four are assumed to be
as above described) by the following illustrations, in
which figures 1 and 2 are reproduced, but with the ulti
mate point in each represented as inaccessible only in a
direct line from the point of beginning, but visible from
such point, and as incapable of being either seen or
reached directly from the third point by reason of an
obstruction in each case, but capable of being reached
indirectly by way of the new point introduced in each
case.

Fig. I a

The sorites are now compound, and fully expressed
are as follows :

In the descending direction.
X comprehends J,
J comprehends D ;

I) comprehends H,
II comprehends N~ \

D comprehends ]> ,
ind (^ . '. X comprehends _N\

and

In the ascending direction.
!N" is comprehended in D,
D is comprehended in J ;

J is comprehended in Q,
Q is comprehended in X j

J is comprehended in X,
,'. N is comprehended in X.

APPENDIX. 121

But a simple sorites may have as its maxinms term a
subaltern genus, and as its magnus term a subaltern
species relatively to new terms which are higher genera
or lower species respectively, and which may be found
by investigation in either or each direction and brought
into the reasoning process in opposite directions re
spectively, and in such case the new terms, if brought in
in both directions, will supplant the original maximus
and magnus terms (or if in one direction only, then
either, as the case may be), and the two latter will be
come major-middle or minor-middle terms, or sub
sidiary middle terms, and the logical significations of the
original middle terms will be changed, the major-middle
becoming minor-middle and the minor-middle subsidiary
in the descending direction, and the minor-middle be
coming major-middle and the major-middle subsidiary
in the ascending direction, or both becoming subsidiary
in either or each direction, according as the number of
new terms brought in in either or each direction shall be
one or two or more than two.

In such a sorites the recognition of the truth of the
premises, and of the necessity of the truth of the ulti
mate conclusion, is assumed as antecedent to further
investigation in either direction.

For the purpose of illustration by geometrical plane
figures, let it be assumed that eight new terms have been
found, four (Y, Z, S, T,) successively in the ascending
direction of the process of investigation, and four (K, Q,
G, H,) successively in the descending direction.

The figures will now be as follows, the original figures
1 and 2 being again reproduced, and two additional

122

APPENDIX.

quadrilaterals annexed to each, four of the points qf
which (two of each quadrilateral) in each case are des
ignated by the symbols of the new terms as above.

Fig. I b

The reasoning process in each direction will now most
naturally (and, necessarily, geometrically considered and
appropriately expressed) fall into the form of a com
pound epicheirema as follows.

Let > signify "comprehends" and < "is com
prehended in."

In the descending direction as in Fig. Ib.

T > S,

8 > X; v S > Z,

and Z > X; V Z > Y,

and Y > X; V Y > X,

and X. > X; Y X > J,
and J > D,
and I) > X;

.-. T > X.

APPENDIX. 123

Or thus :

T > S,
S> Z,
Z > X; v Z > Y,

and Y > X,

and X > X; V X > J,

and J > I),

and D > X;

.-. T >' X.

In the ascending direction ax in Fig. Jb.
H < G,
G < X; v G < Q.

and Q < X: •.• Q < K.

and K < X; '.- K < X^,

and X < X; V X < D,

and ] ) < J,

and J < X;

/. K < X.

Or thus :

II < G,
G < Q,
Q < X: v Q < K.

and K < X.

and X < X: V X < D,

and I) < J,

and J < X;

.-. II < X.

But logically considered, the process may also be in
the form of a compound sorites, in which the premises

124

APPENDIX.

will be found in the order of the lines forming the peri
meter of the figure in each case until the point is reached
which would be the centre of the figure if it were drawn
in the form of a hexagon, and the successive conclusions
are alternate semi-diagonals pointing to the centre, the
last being also the remaining line of the perimeter.

Such compound sorites fully expressed and in both
directions are as follows :

In the descending direction.

T comprehends o,
S comprehends Z ',

Z comprehends i. }
Y comprehends X j

X comprehends J,

( X comprehends ~$$ ,

Z comprehends N,
and .'. T comprehends N.

In the ascending direction.

II is comprehended in G,
Gr is comprehended in Q '

Q is comprehended in K,
L K is coni])rehended in !N \

X is comprehended in I), )
D is comprehended in J, I
J is comprehended in X ;

f ?\ is comprehended in X \

Q is compreliended in X,
and .'. II is comprehended in X.

The transverse diagonals in Figs. 15 and 2b are ana
logues of the unexpressed conclusions of the enthymemes
of the third order into which the first four propositions of
each compound sorites are divided by the first two dotted
lines in each case, and which do not appear subsequently
as expressed premises. The third dotted line in each
compound sorites, with the character .-. prefixed, signi
fies that the conclusion of the premises of the last in-

APPENDIX. 12;")

eluded simple sorites immediately preceding is not ex
pressed as such, but such conclusion follows immediately
as the third premise — in connection with the two premises
between the first and second dotted lines — of the first
included simple sorites ; of which also the conclusion
as such is not expressed, but follows immediately as the
third premise of the new principal simple sorites result
ing from the whole process, composed of the first two
and last two propositions in each case.

The ultimate conclusion of the foregoing compound
sorites in the descending direction is "T comprehends
N," and in the ascending direction is " H is compre
hended in X." If now the two figures be considered as
put together in like manner as Figs. 1 and 2 were put
together to form Fig. 3 (page 16), namely, on their
only common continuous line, J D descending, D J
ascending, it will be found that the relation between the
two extremes T and H, which can be logically demon
strated, cannot be geometrically established by means
of the combined figures.

And if any two of the quadrilaterals, of which Figs. 3
and 19 are composed, are put together on the lines desig
nated by the symbols of the middle terms of the sorites
in each case with additional quadrilaterals annexed to
each, it will be found that the combined figures will not
serve as analogues of the compound sorites demonstrat
ing the relation of the extremes reached by investigation
in both directions.

Let now Figs. lb and 2b be considered as redrawn—
but with the original quadrilaterals in such form that
each whole figure shall be exteriorly hexagonal — and

126

APPENDIX.

put together on their common line as in the following
figure :

The term of beginning of each original sorites is at
an acute angle of the quadrilateral by which the sorites
is represented instead of at an obtuse angle as in all pre
vious figures, and the diagonal representing the unex
pressed conclusion of the first which is the unexpressed
premise of the second of the two syllogisms into which
each sorites may be expanded is transverse relatively to
that shown in previous figures, but which does not
appear in these.

The figures considered separately or as combined are
now in such form that they are perfect analogues of the
compound processes of reasoning in so far as such pro
cesses may be exhibited on regular plane figures.

APPENDIX. 127

The premises of such process in the descending direc
tion (on the combined figures) begin with T and follow
the perimeter of the upper half of the figure until the
point X is reached, and in the ascending direction begin
with H and follow the perimeter of the lower half of the
figure until the point N is reached, and then in each
case follow the transverse diagonal in the original quad
rilateral and continue thence along the whole perimeter
of the other half of the figure until the ultimate term is
reached ; and the successive conclusions are represented
in the last half of the figure (first returning) by two
transverse diagonals forming an angle, the vertex of
which is at the point H in the descending direction and
at the point T in the ascending — thence, per saltum,
from the term of beginning of the original sorites in one
direction to the term of beginning of the original sorites
in the other, by means of the conclusion assumed to have
been found in the original sorites in each case (but not
•expressed in the compound process) along a diagonal
not shown in the figure connecting the terms of begin
ning of the two original sorites on the two original
quadrilaterals — and are then further represented in the
first half of the figure (last returning) by the line of its
perimeter which does not represent a premise, then by a
diagonal of the first half of the figure (of which diagonal
only half is shown in the figure, but the arrow-head
therein points to the ultimate term\ and lastly by the
line representing the original middle premise in both
directions being the line common to each half of the
figure.

128 APPENDIX.

The compound sorites thus described, fully expressed
in both directions, are as follows :

In the descending direction. In the ascending direction.

T comprehends S,
S comprehends Z ;

Z comprehends Y,

[_ \ comprehends X 5

^ X comprehends I),
/ D comprehends N J

{X comprehends K,

K comprehends Q 5

Q comprehends Gr, 1
Gr comprehends II ', |

<•

Q comprehends II 5
1 .'.-

N comprehends II :

I 1

X comprehends II '

Z comprehends H,
and .'. T comprehends H.

II is comprehended in Gf,

Gr is comprehended in Q '

Q is comprehended in K,

K is comprehended in X \

j N is comprehended in J,

( J is comprehended in X j

X is comprehended in Y,

\ is comprehended in / j

Z is comprehended in S, 1

S is comprehended in T J f

Z is comprehended in T :

X is comprehended in T j

X is comprehended in T J

Q is comprehended in T,

and .'. II is comprehended in T.

The j)rincipal simple sorites resulting from each of
the foregoing compound sorites are not correlatives of
tiach other, although reaching the same ultimate conclu
sion, or rather conclusions which are convertible into
each other. But the typical simple sorites, hereinbefore
described, as represented in the figure (not in the sorites)
would be as follows (progressive and regressive in each
direction), and are correlatives of each other :

APPENDIX. 129

Progressive descending. Regressive ascending.

T comprehends J, J is comprehended in T,

J comprehends U, D is comprehended in J,

1) comprehends H ; II is comprehended in 13 j

.". T comprehends H. .*. II is comprehended in T.

Progressive ascending. Regressive descending.

II ie comprehended in D, D comprehends H,

D is comprehended in J, J comprehends D,

J is comprehended in Tj T comprehends J'

.". II ie comprehended in T. .'. T comprehends II.

The typical simple sorites is represented by one and
the same line in the figure, the first premise in each (pro
gressive in each direction) being represented by such line,
considered as designated by a symbol without and a sym
bol within the figure, the second and middle premise by
the two symbols within the figure, the third premise by
a symbol within and a symbol without the figure, and
the conclusion by the two symbols without the figure.

But the typical simple sorites as represented in the
compound sorites (not in the figure) would be as follows :

Progressive descending Regressive ascending.

T comprehends X, X is comprehended in T,

X comprehends N, N is comprehended in X,

X comprehends H; H is comprehended in N°

.'. T comprehends H. .*. H is comprehended in T.

Progressive ascending. Regressive descending.

H is comprehended in N, N comprehends H,

N" is comprehended in X, X comprehends N,

X is comprehended in Tj T comprehends Xj

.*. H is comprehended in T, .', T comprehends H.

130 APPENDIX.

It will now be remembered that the original processes
in both directions were assumed as having been gone
through with, and their respective ultimate conclusions
established antecedently to further investigation in either
direction. The terms of such conclusions may therefore
be considered as representing a concrete genus and a
concrete species proximate to each other, or so nearly so
that the relation of each to the other is recognized and
admitted. Thus the two forms of the typical simple
sorites represented by, and taken from, both the figure
and the compound sorites, are justified.

Let now the parallelogram represented in the figure
by triangles 1 and 3 be considered as taken out, and let
the two remaining parts of the figure be considered as
put together on the lines of the two transverse diagonals
in the original quadrilaterals as common to both.

The two premises in each of the compound sorites,
namely, "X comprehends D" and "D comprehends ]N" "
in the descending direction, and "N is comprehended in
J" and "J is comprehended in X" in the ascending
direction, will now be supplanted by their two conclu
sions respectively which would read thus, "But X com
prehends N" and "But N is comprehended in X," and
let the word "and " be prefixed to the next proposition
in each case. The typical simple sorites above given, as
taken from the two compound sorites as thus changed in
form, are thereby further justified. But the new figure
and the compound sorites will both be irregular in form,
the latter consisting of the premises of one process in
each case conjoined — by the proposition thus substi
tuted—to the premises of another process originally in

APPENDIX. 131

the opposite direction, but the direction changed so that
the whole process with the successive conclusions fol
lowing may be in one and the same direction.

The typical regressive simple sorites in both forms
may be reduced to the form of simple syllogisms as
follows :

As taken from the figure,

In the descending direction. In the ascending direction.

3 ) comprehends H, J is comprehended in I ,

T comprehends D J His comprehended in J ;

.'.- T comprehends H. .% H is comprehended in T.

As taken from the compound sorites,

In the descending direction. In the ascending direction.

^ comprehends H, X is comprehended in T,

T comprehends !N J His comprehended in X ;

.'. _T comprehends II. .'. II is comprehended in T.

The premises of these syllogisms in each case are not
correlatives of each other, although the conclusions are,
and it is only by means of the simple sorites, in the
middle premise of which the two middle terms of such
syllogisms in each case are compared, that the complete
correlation of the processes of reasoning in both direc
tions can be exhibited. The claim of the simple sorites,
singly or two combined, to be regarded as the complete
and (considered as combined) necessary form of the
process of reasoning is thus vindicated.

Thus far only can the analogy between compound
logical processes and combinations of regular geometrical
plane figures be exhibited on paper, but the process in

132 APPENDIX.

respect to each direction of the process of investigation
may be considered as continued indefinitely in a circular
direction about the central points in Figs. Ib and 25,
either as originally drawn or as redrawn ; but if investi
gation shall have been made in both directions, then such
investigation and the reasoning processes in respect
thereto, in so- far as they may be represented by a com
bination of regular geometrical figures, are limited as
shown in the foregoing figure.

But on combined irregular figures, as before described,
with other like figures annexed laterally in both direc
tions (proceeding upwardly to the right and downwardly
to the left), the compound processes may be logically
pursued indefinitely.

If the original quadrilaterals in Figs. \b and 25 had
been in the form of squares, as in Figs. 4 and 5 on page 17,
the annexed quadrilaterals would have been also in like
form, but there would have been three in each figure in
stead of two as in Figs. 15 and 26, and the figures would
have been perfect analogues of compound processes of rea
soning in each direction, in like manner as Figs. 15 and 25,
with two additional terms brought in in each direction.

But if the two figures are considered as put together on
the lines J D and D J, it will be found that the resulting
figure is not a perfect analogue of the compound reason
ing processes establishing the relation of the two extremes
reached by investigation in both directions. The conclu
sions of two of the included sorites would not be repre
sented in the first half of the figure (last returning) in each
direction by lines of the figure, but would be reached by
indirection. Thus the square in respect to such combined
figures is an imperfect analogue of the sorites.

APPENDIX. 133

The accompanying figures, 3& and 19& (on a folded
sheet following page 134), exhibit the original combina
tions of quadrilaterals, as in Figs. 3 (page 16) and 19
(page 35), considered as the faces of tetrahedra of edge
1 spread out, with five additional combinations annexed
to each, five new terms being assumed to have been
found by investigation in each direction, two in each of
the first two annexed combinations and one in the third,
the latter being found in the third annexed combination
to be related to the fourth term of the sorites on the faces
of the original combination, but not at the same point,
both the original complete processes of reasoning being
assumed to have been gone through with and their re
spective ultimate conclusions established, antecedently
to further investigation in either direction. The course
of the process of investigation will now be found by an
examination of the figures to have been ascending from
X, in Fig. 35, along the lines which are analogues of the
premises of the syllogisms represented by the several tri
angles marked 3 ; and descending from N', in Fig. 19£>,
along the similar lines of the triangles marked <?, until in
the first case the ultimate highest point X' was reached,
and in the second case the ultimate lowest point N.

The compound processes of reasoning, retracing these
lines respectively, descend in the first case from X' to X,
and thence, along the similar lines of triangles 1 and 3 in
the original combination, to the ultimate point IST, and
in the second case ascend in like manner from N to N',
and thence, along the similar lines of triangles a and c
in the original combination, to the ultimate point X'.
Such processes constitute the following compound sorites
fully expressed in both directions.

134

APPENDIX.

In the descending direction
as in fig. 3b.

In the ascending direction
as in fig. 19b.

X' comprehends D ,

Dl • V'
comprehends ^N 5

X is comprehended in J,
J is comprehended in X ;

{N"' comprehends J?
J comprehends K";

1 X is comprehended in I)',
[^ l) is comprehended in X^ 5

]V comprehends P,
1 P comprehends Xj

X' is comprehended in B,
1 13 is comprehended in II 5

{X comprehends S.
tS comprehends Z ;

f Z comprehends j. ,
Y comprehends X ;

f II is comprehended in Gr,
i G is comprehended in Qj

j' Q is comprehended in K,
K is comprehended in X';

X comprehends J, "^

•f T Tk

J comjirehends l)f
I) comprehends X;

C X comprehends !N ;
r Z comprehends !N '.
r X comprehends X;

I
X' is comprehended in D',

D' is comprehended in J'.
J' is comprehended in X';

C ^N is comprehended in X 5
r Q is comprehended in X' ;
C 11 is comprehended in Xj

C ]N comprehends X;

f X is comprehended in X J

X' comprehends X,
and .'. X' comprehends X.

X is comprehended in X ,
and .'. Js is comprehended in A .

Fig. 3 b

X' D'/\D' X

134

Fig. 19 b

APPENDIX. 135

Two of the unexpressed conclusions, but expressed as
premises (one in each compound sorites), namely, "N
comprehends N" in the descending direction, and "X'
is comprehended in X'" in the ascending direction, have
the symbols designating their terms alike in each case, but
by reference to the figures such symbols will be found to
designate two different points in each case, and therefore
two different terms. These symbols, where they designate
exterior points of the oct'astron respectively, are put in
full-faced type in the figures and also in the sorites.

But the process of reasoning may be exhibited in a
shorter form in each case, than by retracing the lines of
the process of investigation, namely, as follows :

In the descending direction. In the ascending direction.

X' comprehends X', N is comprehended in X,

K"' comprehends N, X is comprehended in X',

X comprehends T, X' is comprehended in H,

T comprehends Z, H is comprehended in Q,

Z comprehends X J Q is comprehended in !N J

.*. X' comprehends X. . ". X is comprehended in N'.

But X comprehends N, But X' is comprehended in X',

and .'. X' comprehends N. and .'. X is comprehended in X'.

Each process is in the form of a compound sorites,
not fully expressed, to which is appended an enthymeme
consisting of the ultimate conclusion of the original prin
cipal sorites as a premise, and the conclusion resulting
therefrom and the preceding conclusion of the compound
sorites considered as the suppressed premise of the
enthymeme.

The analogues of the premises of the compound sorites

136 APPENDIX.

in each case will be found to consist of the dotted diag
onals, by which one of the quadrilaterals of each annexed
combination is divided into triangles 3 and 4, in Fig. 3&,
and c and d, in Fig. 19&, and which are also analogues
of the conclusions of the processes on triangles 3 and c
respectively considered as conducted pari passu with
the process of investigation, and the ultimate conclusions
of the compound sorites, " X' comprehends X" descend
ing, and " N is comprehended in N" ' ascending, are
represented in the figures by all such dotted diagonals
in each figure respectively, forming in each case one and
the same straight line connecting the ultimate point
reached with the point of beginning of investigation.

The process of investigation in Fig. 35 is ascending,
and in Fig. 19# descending, and in strict accordance with
the forms of logic the processes of reasoning should have
been in the regressive configuration in the same directions
respectively (into which the processes as shown are con
vertible — see page 16), the process of investigation toeing
always progressive, and that of reasoning in retracing the
steps regressive, but in such case the compound process,
when it should reach the principal combination, would
require that the direction of the original process, assumed
to have been gone through with thereon, should be
changed, or otherwise there would be in each case a com
bination of processes in opposite directions.

If now the figures be drawn upon and cut from card
board, and the board cut half-way through on all the
lines (on the interior lines of each combination, on the
face of the figure as shown, but on the lines connecting
the combinations, on the other side), each of the figures.

APPENDIX.

137

may be folded so as to inclose six volumes of space, each
in the form of a regular tetrahedron, connected each two
by an edge in such manner that the line of each con
necting edge shall be perpendicular (on both sides) to a
plane in which the opposite edges of the two connected
tetrahedra shall lie.

The figures cut and folded as described will be found
to take the form shown in the following illustration :

And let the edges of the first and last tetrahedra, which
will come together, be fastened together.

Such edges in the descending direction (Fig. 3&) are
D N, common to triangles 2 and 3 of the original combi
nation of quadrilaterals (when folded), and X' N, com
mon to triangles 2 and 4 of the last annexed combina
tion ; and in the ascending direction (Fig. 19&) are J' X',
common to triangles b and c of the original combination,
and N X', common to triangles b and d of the last an
nexed combination. The outer point, common to the
two edges brought together in the folding of each figure,
is designated by two symbols, viz. : D and X' in the
descending direction and J' and N in the ascending.

138 APPENDIX.

In like manner, as shown in respect to the quadri
laterals considered as plane figures, the combined and
folded quadrilaterals have each made a complete circuit,
and the analogy between the logical processes and geo
metrical solid figures cannot be further exhibited, but
the processes may be considered as further indefinitely
continued proceeding successively along the lines of the
surfaces of the same volumes of space. But in plane
figures exhibiting the faces of the solid figures spread
out they may be represented on paper indefinitely.

The circuit thus formed in each case is similar, in re
spect to the relative positions to each other of the tetra-
hedra of which it consists, to six of the eight tetrahedra
of which with six octahedra an octahedron of edge 2
is composed, in which considered as contained within
a right triangular pyramid of edge 4, the centre of-
the pyramid may be reached as described on pages 62
and 63.

The exterior face of the original tetrahedron in cir
cuit 35 is triangle 1, and in circuit 19& is triangle «, and
that of each of the annexed tetrahedra in the former is
triangle 3 and in the latter triangle c. Thus the courses
of the processes of investigation and reasoning along the
lines of such faces until the original tetrahedron is
reached returning (on the faces of which the processes
were assumed as gone through with antecedently to fur
ther investigation) are wholly on the surface, the lines of
the shorter processes being the boundaries in each case
of the hexagon composed of the interior faces of the
tetrahedra of the circuit opposite their exterior vertices
respectively.

APPENDIX.

139

Let now the eight superposed tetrahedra of the oct'-
astron be considered each as named by the symbol at its
exterior vertex, as in Figs. 17 and 18 (page 34), and let
a card-board figure of the included octahedron be con
sidered as in hand together with the two circuits of
card-board tetrahedra formed by the folding of the fore
going figures.

If now the two circuits of tetrahedra be applied to
the octahedron in such manner that the first and principal
tetrahedron of each formed by the folding of the original
combination of quadrilaterals shall occupy the positions
respectively of tetrahedra X and N' relatively to each
other, as in the oct'astron held as shown in Fig. 17,
they will be found, circuit 36 descending backwardly,
but obliquely, to the right, and circuit 196 ascending
forwardly and obliquely to the left, the second, third,
and fourth tetrahedra in each circuit being entirely out
side of the oct'astron, and the fifth and sixth in circuit
36 being N and D', and in circuit 196 X' and J. Six
faces of the octahedron will have been covered, leaving
only those exposed on which in the completed oct astron
tetrahedra D and J' are superposed.

The principal simple sorites resulting from the two
fully expressed compound sorites, on page 134, are as
follows :

hi the descending direction.
\ comprehends D',
1 ) comprehends N ,
J\ comprehends N I

.*. X comprehend-; X.

In the ascending direction.
TS is comprehended in J,
.1 is comprehended in A.,
X is comprehended in X';

. ^N is comprehended in X .

And the principal simple syllogisms resulting from the

140 APPENDIX.

shorter processes on page 135, in which the successive
conclusions of the reasoning process conducted part
passu with the process of investigation are employed
as premises, are as follows :

In the di'xci'itdiny direction. In the ascending direction.

X' comprehends X, X is comprehended in X',

X comprehends X; X' is comprehended in X';

.'. X' comprehends X. .'. X is comprehended in X'.

Thus, from two diametrically opposite stand-points,
as shown by the two circuits applied to the octrahedron,
and by paths entirely diverse, the same ultimate result
is reached, but expressed, in the one case descending
from above, as the greater comprehending the less, and
in the other case, ascending from beneath, as the less
comprehended in the greater.

If now the two complete processes be considered
each as reversed in direction, so that the first, instead
of descending from X at the upper pole of the axis of
revolution to N" on the lower horizontal line (see Fig. 18,
page 34), shall ascend from N to X, and the second,
instead of ascending from N' at the lower pole of such
axis to X' on the upper horizontal line (see Fig. 17, page
34), shall descend from X' to N', and if figures should
be drawn showing the processes so conducted, such fig
ures would be appropriately numbered Figs. 3c and 19c,
and would be similar to Figs. 3& and 19Z> except that the
original combination would be the upper combination in
the former and the lower in the latter, and the annexed
combinations in each case would proceed to the left in
stead of to the right. The faces of the principal and first

APPENDIX. 141

annexed combinations which would adjoin each other
would be 4 and 3 in Fig. 3c, instead of 2 and 1 as in Fig.
35, and d and c in Fig. 19c, instead of b and a as in
Fig. 195. The symbols, except the last, employed in the
first three annexed combinations in Fig. 3c, may be those
of Fig. 195, and in Fig. 19c those of Fig. 35, but marked
as primes in each case, their employment as such serving.
to show their significations as comprehending or compre
hended as they were originally employed. Fig. 3c thus
becomes an ascending circuit and Fig. 19c descending.

Such figures could be folded in the form of circuits of
tetrahedra and applied to the faces of an octahedron
similarly to Figs. 35 and 195, but the tetrahedra formed
by the folding of the original combinations would oc
cupy the positions of tetrahedra IN" and X' of the oct'as-
tron. By such application the faces of the octahedron
left exposed, as described on page 139, would be covered
and instead there would be left exposed the faces on
which in the completed oct' astron tetrahedra J and D'
are superposed. The exterior faces of the circuits, in
stead of being 1 and 3 as in Fig. 35, and a and c as in Fig.
195, as described on page 138, would be 3 and 1 in Fig. 3c
and c and a in Fig. 19c.

The principal simple sorites which would result from
the compound sorites in such case would be as follows :

In the ascending circuit as would In the descending circuit as would

be shown in Fig. 3c. be shown in Fig. 19c.

N"' is comprehended in J', X comprehends D,

J' is comprehended in X , D comprehends N,

X' is comprehended in X ; N comprehends ^NT'j

,' . N' is comprehended in X. .'. X comprehends JST'.

142 APPENDIX.

And the principal simple syllogisms which would result
from the shorter compound processes would be as follows :

In the ascending circuit. In the descending circuit.

N' is comprehended in N", X comprehends X',

N is comprehended in Xj X' comprehends X'j

.*. X ' is comprehended in X. /. X comprehends X'.

Thus the relations to each other of like extremes of
each of the two complete processes of reasoning on the
faces of the oct'astron are established, namely, of X and
X' (the original ultimate terms) to each other as in the
processes previously shown, and of X and N' (the original
terms of beginning, but ultimate in the reverse directions)
to each other as in the processes of which the results in
simple form have been just shown, and by examining
the principal simple sorites resulting from the compound
sorites in each case it will be seen that the combination
of the two complete processes of reasoning in one har
monious whole is accomplished in each direction through
the third term in conjunction with the fourth in the same
direction, being the second and first in the opposite direc
tion, as middle terms, and all other terms brought in in
the full processes are subsidiary.

But the two complete processes of reasoning of the
like extremes of which the relations to each other are
thus established are not those on the faces of the oct'as
tron considered as consisting of two intervolved tetra-
hedra, but as consisting of eight tetrahedra superposed
on the faces of an octahedron and having their points
(that is, all the points of each) designated by the logical
symbols similarly to those of the intervolved tetrahedra,
and so superposed relatively to each other that the whole

APPENDIX. 148

figure shall have its exterior points designated similarly
to the oct'asiron considered as consisting of two inter-
volved tetrahedra, us described in § 11 on page 37.

By examining the tetrahedra of the circuits 3b and
I9b, it will be found that the new principal simple sorites
resulting from the two compound sorites are the simple
sorites represented on the faces of the last annexed tetra
hedron in each case, and in like manner it would be found
in respect to circuits 3c and 19c if those figures were
drawn and folded as hereinbefore described, and the
compound sorites represented thereon fully expressed.

Mathematically considered, each link of the chain of
reasoning is represented by a triangle and is limited to
three terms, but logically considered each link is repre
sented by a combination of quadrilaterals on which the
reasoning is exhibited as both descending and ascend
ing (but expressed only in either direction), and extends,
but is limited to four terms.

Let nowT the superposed tetrahedra be considered
each as having its exterior point designated by the sym
bol which is the term of beginning of the combined so
rites of which it is the analogue, namely, the four which
together, with the included octahedron, constitute the in-
tervolved tetrahedron in the descending direction, each
by the symbol X, and the four in the ascending direc
tion each by N'. It will now be necessary to add a
separate appellation to each of such terms other than the
two originals, so as to distinguish them from the origi
nals respectively and from each other, which appellations
it will be necessary also to add to each of the other terms.

Let the following be the full symbols of the terms in.

144 APPENDIX.

their regular order in each case, the column at the left
hand showing the designation of the first term on the
faces of the intervolved tetrahedra of the oct'astron :

Of the tetrahedra considered Of the tetrahedra considered

as descending. as ascending.

X J D N N'

X0 J, D, X, D'

X4 J4 D4 X4 J'

X6 J6 D6 X6 X'

N' D' J' X'

JN 3 1^3 J , Xo

^N g Dg J 5 Xg

N, D, J7 X7

If now figures were drawn corresponding to Figs. 35
and 196 (and which would be appropriately numbered
Figs. 3d and 19(2), it is manifest that the processes of in
vestigation and reasoning along the lines of the first
three consecutively annexed combinations of quadrilat
erals in such figures and the designations of the points
of such quadrilaterals (except of one point in the third
annexed combination in each case, which will have to be
changed, as will be hereinafter shown) would be the
same as in Figs. 35 and 195.

But when the fourth annexed combination is reached
in each case it will be found that although the physical
construction of the circuit may be the same, and spread
out in plane figures as in Figs. 35 and 195, it will be
necessary to change the order of the faces as to their
numbered or lettered designations, so that the exterior
points of the tetrahedra formed by the folding of such
annexed combination in each case shall be X6 instead of
N, and N7 instead of X', as in the oct'astron considered
as consisting of intervolved tetrahedra, and as shown in
the figures, and accordingly the face of the fourth an-

APPENDIX.

145

nexed combination adjacent to face 4 of the third must
be face 2, in Fig. 3d, instead of face 1 as in Fig. 36, and
the face adjacent to face d must be face 6 in Fig. 19d,
instead of face a as in Fig. 196. These changes will now
require the substitution of X6 in place of N, in the third
annexed combination in Fig. 3d, and of N7 in place of
X', in the like combination in Fig. 19d, and the points
of the fourth annexed combination in Fig. '3d will be

designated X6, J6, D6, and N6, and in Fig. 19^.N7, D7,
J7, and X7.

The fourth annexed combination in the two figures
will now be as follows :

In Fig. 3d.

In Fig. 19d.

These figures being considered as substituted in
Figs. 36 arid 196, and the symbol designating one of the
points in the third annexed combination of quadrilaterals
in each case changed as. above described, and all the
symbols in the last combination obliterated, the figures
may be folded as before and the geometrical processes
may be pursued if the symbols be regarded as having
no signification other than as designating the points,
but if they are considered as retaining their logical sig-

146 APPENDIX.

nifications, then all reasoning is at an end when the
fourth annexed combination is reached.

It will be necessary only to consider the syllogisms on
face 2 in Fig. 3d and b in Fig. IVd. They may be con
sidered in two ways ; first, in the order in which they
have been hitherto considered as proceeding (but on
face 1 in 3b and a in 19&), in which case one of the
premises will be found to be untrue although the con
clusion will be true, and, secondly, in the order in which
the premises are both true, but the conclusion (all the
propositions being, logically considered, universal-affirm
ative), although still true, .will be found to be unwar
ranted.

Thus, in the order in which they have been hitherto
considered as proceeding :

On face 2, in Fig. 3d. On face b, in Fig. 19d.

X g comprehends Xg, ^i i8 comprehended iu X-,

Dg comprehends X6; Jij is comprehended in ^N ^ ;

.'. D6 comprehends Ng. .'. J-j is comprehended in X,.

Here the second premise in each syllogism is untrue,
although the conclusion in each is true.

And, secondly, in the order in which the premises
are both true :

On face 2, in Fig. 3d. ' On face b, in Fig. 19d.

X6 comprehends Ng, Ni, is comprehended in X7,

Xg comprehends Dg; N, is comprehended in J7;

.'. D6 comprehends Ng. .'. J7 is comprehended in X,.

Here the syllogism in each case is in the third figure

APPENDIX. 147

of logic in which only a particular conclusion can be de
duced, which is not the case in either of the syllogisms
as stated. The conclusions, therefore, although still
true, are unwarranted.

Thus it will be seen that the two complete processes
of reasoning on the faces of an oct'astron, considered as
consisting of tetrahedra superposed upon an octahedron,
cannot be linked together when the exterior points of
the oct'astron are designated by the terms of beginning
of the processes respectively, but only when designated
similarly to the points of the oct' astron considered as con
sisting of two intervolved tetrahedra ; and, on the other
hand, the two processes on the faces of the intervolved
tetrahedra cannot be linked together so that the relations
of their extremes designating opposite poles of each of
two axes of the oct'astron shall be established. But they
can be so linked together on the surface of the circum
scribed sphere of the oct'astron considered as consisting
of both intervolved and superposed tetrahedra, as will be
herein next shown. So wonderfully and perfectly is nature
consistent with herself throughout her whole domain.

The sphere as described in § 18 (on page 109) et seq.y
is the circumscribed sphere of the oct'astron considered
as consisting of two intervolved tetrahedra.

Let it now be considered as the circumscribed sphere
of the oct'astron consisting also of tetrahedra superposed
upon the included octahedron, as described in § 11, on
page 37.

The exterior points of the oct'astron so considered
will be designated on the surface of the sphere in like
manner as before — see Figs. 34 and 35, on pages 111

148 APPENDIX.

and 112 — but those points of the tetrahedra considered
as superposed which fall upon the included octahedron
do not come to the surface of the sphere, and their desig
nations will not, therefore, appear, unless brought to the
surface through some of the figures thereon, which, it
will be observed, are not figures of the oct'astron, but of
curved sections of the added figures in the construction
of the octahedron of the second order. They may ac
cordingly and appropriately be considered as brought to
the surface through the rhombs at their obtuse angles,
and Figs. 34 and 35 are herewith reproduced with such
obtuse angles so designated, except that Fig. 35 is ex
hibited as held above the eye, as described in the text
following that figure.

Both the figures are considered as representing the
sphere held with its axis X N' vertical in such position
below the eye in Fig. 34a that the line of vision shall
be perpendicular to the surface at the point X' and shall
coincide with the axis X' 1ST, and above the eye in Fig. 3oa
that such line shall be perpendicular to the surface at the
point N and shall coincide with the same axis N X'.

The syllogism on face 1 of each of the three tetrahe
dra named X, J, and D, and the three syllogisms on faces
c, b, and d of the tetrahedron named X' are brought to
the surface in Fig. 34a, and the syllogism on face a of
each of the three tetrahedra named N', D', and J', and the
three syllogisms on faces 3, 2, and 4 of the tetrahedron
named N are brought to the surface in Fig. 35a.

It will be necessary only to consider such syllogisms on
the upper rhomb in Fig. 34« and the lower one in Fig. 35a.
They are considered, those in Fig. 34a both as descend-

APPENDIX.

149

Fig. 35a

150 APPENDIX.

ing and those in Fig. 3o« both as ascending, accordingly
as they are viewed from above or below the sphere.
They are as follows :

In Fig. 34u.

X comprehends J} X' comprehends J',

J comprehends D, J' comprehends D',

.'. X comprehends D. .'. X comprehends 13 .

In Fig. 35a.

N' is comprehended in D', X is comprehended in 13,

I) is comprehended in J } 13 is comprehended in (J }

.'. X is comprehended in J'. .'. JS is comprehended in J.

The conclusion of each of these syllogisms establishes
the relation between the first and third terms of the com
bined sorites descending or ascending throughout on the
faces of each superposed tetrahedron considered as a
whole figure.

Let now such conclusion in each case be taken as the
first premise of the syllogism represented by one of the
triangles formed by the transverse diagonal in each case
in like manner as in the figure on page 126 representing
the sorites as exhibited on plane geometrical figures, and
there will be found the following four syllogisms :

In Fig. 3Jtu.

X comprehends 13, X comprehends D',

D comprehends X', D' comprehends X,

.'. X comprehends X'. .*. X' comprehends X.

In Fig. 35a.

N' is comprehended in J'. N" is comprehended in J,

J is comprehended in N, J is comprehended in X,

.'. N' is comprehended in X. .'. X is comprehended in N'.

APPENDIX. 151

Thus the equality to each other of the maximus terms
of both processes is established in Fig. 34«, and in like
manner that of the magnus terms in Fig. 35a.

Let now the foregoing syllogisms considered as the
ultimate results of the processes on the faces of the
superposed tetrahedra be combined with the conclusions
of the processes on the faces of the intervolved tetrahedra
and the resulting conclusions will be found to establish
the relations of the terms by which opposite poles of each
of two axes of the oct astron, and of its circumscribed
sphere are designated.

Such combinations of processes will be as follows :

In Fig. 3J^a.

X comprehends D, X' comprehends D ,

D comprehends X , D' comprehends X,

.*. A comprehends X . .".X comprehends X.

But X' comprehends If , But X comprehends X,

and .'. X comprehends N . and .'. X comprehends .N.

In Fig. S5a.

K" is comprehended iu J', X is comprehended in J,

J is comprehended in N" , J is comprehended in N

.'. X' is comprehended in X. .'. X is comprehended in N'

But X is comprehended in X, But X' is comprehended in X',

and .'. X' is comprehended in X. and .'. X is comprehended in X.

All the foregoing syllogisms, as indeed also all the
syllogisms previously exhibited except those which are
regressive, are in the fourth figure of logic and it may be
^objected that the conclusions are therefore unwarranted.

152 APPENDIX.

But all the terms, whether employed as subject or predi
cate, are herein considered as distributed, and the fourth
figure in such case is the natural and therefore perfect
figure of logic. But the premises, in order to make the
syllogisms conform to the first figure of logic, may be
considered as transposed, in which case all the syllo
gisms will be regressive.

To return now to the consideration of the circuits.
They have been considered as applied to the octahedron
in pairs, 3b and 19& together and 3c and 19c together, in
each of which cases each circuit is wholly independent of
the other ; and in the case of each circuit there are three
tetrahedra entirely without the oct'astron considered as
complete.

Let them now be considered as applied to the octa
hedron in pairs, as follows : 3b and 3c together and I9b
and 19c together.

It will now be found that the two circuits in each
case are not independent of each other, but are interde
pendent, four of the tetrahedra of each being common to
both, namely, two of the tetrahedra which are tetrahedra
of the oct'astron — in the case of 35 and 3c X and N (so
named) descending and N and X ascending, and in the
case of 19& and 19c W and X' ascending and X' and N'
descending — and two of the tetrahedra in each case which
are without the oct'astron, namely, those which are con
nected by their edges with the above-named tetrahedra
being the first and third of the outside tetrahedra (but
second and fourth of the circuit) in each direction. The
third of the tetrahedra of the circuits which are also
tetrahedra of the oct'astron (sixth of the circuits) are, as.

APPENDIX. 153

has been before seen in the case of 35, the tetrahedron
named D' and in the case of 195, J, and upon examina
tion of 3c would be found to be J', and in the case of 19c,
D ; and the second of each of the outside tetrahedra in
each circuit (third of the circuit) is different from the
second in the other.

Thus, by the application of the four circuits to the
octahedron, all the faces of the latter have been covered,
and the two intervolved tetrahedra of the oct'astron are
complete with four outside tetrahedra annexed to each,
making eight on the two conjoined and considered as
descending from X and ascending from N', the poles of
one axis, and as ascending from N and descending from
X', the poles of another axis.

The combinations of the interdependent circuits, 35
and 3c together, and 195 and 19c together, consist each
of the eight tetrahedra, of which with six octahedra an
octahedron of edge 2 is composed, as described on pages
62 and 63 and referred to on page 138.

The centre of each combination is common to the two
circuits of the combination and is designated by the sym
bols of the ultimate terms of the two processes on the
faces of the two principal superposed tetrahedra, the
first of each circuit.

The eight outside tetrahedra will be found by a careful
examination of the circuits to be connected by their edges
(one of each) with the eight tetrahedra of the oct'astron
by one of their exterior edges respectively, namely, those
in circuits 35 and 3c with tetrahedra X, J', N, and D' and
those in circuits 195 and 19c with tetrahedra N', D, X',
and J. The edges of the intervolved tetrahedra which are

154 APPENDIX.

thus connected with the outside tetrahedra in the two
interdependent circuits 3b and 3c are X N and J' D', and
in the two interdependent circuits 195 and 19c N' X7 and
D J. If the oct'astron be considered as the nucleus of a
cube, such edges in pairs as above will be found to be
the diagonals of two opposite faces of the cube.

But there are three exterior edges of each of the
superposed tetrahedra of the oct'astron, and it will now
be manifest that if the oct'astron be approached on either
one of its other two sides, faces 2 and 4 descending, or b
and d ascending, the side on which it is approached may
have the points of its face designated by the symbols of
triangle 1 or a (as the case may be) and the symbols of
the other triangles (in both directions) will take their ap
propriate places on the other faces accordingly. The
symbols of the terms of beginning of the two complete
processes will be at the same points, but considered as
the stand-points from which the processes are to be con
ducted, on different faces in each case from those in
which they have been hitherto considered, but the sym
bols of all the other terms will not only be on different
faces, but also at different points. The processes being
now considered as conducted from such stand -points suc
cessively, the outside tetrahedra of the new circuits
which would be formed would be connected with the
tetrahedra of the oct'astron by edges of the latter respect
ively different in each case from those before described,
beginning with the edge opposite the face on which the
reasoning is considered as beginning, thus bringing the
number of such outside tetrahedra up to twenty-four.

The whole figure formed by all the circuits consists

APPENDIX. 155

of the oct'astron and the twenty-four tetrahedra men
tioned in the description of the construction of an octa
hedron of the second order on page 51, and may be called
the skeleton of such octahedron.

By the application of all the circuits to the octahe
dron their centres are found, N in 3b and X in 3c at the
middle point of edge X N of the intervolved tetrahedron
in the descending direction, and X' in 195 and N' in We
at the middle point of edge N' X' of the intervolved
tetrahedron in the ascending direction, such middle
points being opposite poles — in each case of processes
Conducted from different stand-points — of one of the
three axes of the octahedron. Thus it will be seen that
the points of the octahedron can only be considered as
designated by the symbols of the ultimate extremes of
the two processes.

But two of the points only, in each case, and not the
centre, of the octahedron have been reached.

Comparing the cube as it was considered with refer
ence to its construction on pages 10 and 44, with the
oct'astron and the circuits of tetrahedra, it will be ob
served that it differs from each of them in this impor
tant respect, namely, that in the oct'astron the faces of
the tetrahedra are superposed upon an octahedron and
in the circuits may be considered as so superposed, the
volumes of space left between the tetrahedra of the
circuits and all outer space, in all cases inviting the in
terposition therein of other octahedra (all the circuits
being considered as having been gone through with),
with the evident design on the part- of nature that their
outer faces can be in like manner proceeded along and

156 APPENDIX.

built upon, but in the cube (edge 1) the faces of its in
cluded tetrahedron or (edge 2) of the tetrahedra of its
included oct'astron have superposed thereon other and
irregular tetrahedra, there being, in the latter case, but
one octahedron, the points of which, only, come to the sur
face at the centres of the faces of the cube, and the fig
ures which can thereafter be superposed upon the cube are
only those similar to the fully completed constructure.

The octahedra thus invited to be interposed in the
circuits, being considered as in fact interposed, the entire
figure is an octahedron of the second order, on the faces
of which tetrahedra of three times the edge of the original
superposed tetrahedra may be superposed and considered
as the analogues of wider processes of reasoning, and the
building up of the figure and the processes of reasoning
may be in like manner continued indefinitely, widening
as they progress throughout all conceivable regions of
space.

The author closes (as both logically and geometrically
he should, returning to the point of beginning) by re
curring to the main question concerning which it has
just occurred to him that it has nowhere throughout the
book been formulated as an interrogatory, and that if
so formulated, it could be enlarged and put forth as
follows :

Would the processes of geometry be in any wise
affected by changing the forms of the units of measure
of surface and solidity, and if yea, then how, in respect
to each possible change to regular figures, favorably or
unfavorably ?

And it seems to him that put in this form it carries

APPENDIX. 157

with it conviction of the necessity of an affirmative
answer to the main question as herein first stated, as it
presents immediately the antithesis of change from the
square and cube, both regular figures, to the regular
triangle and tetrahedron on the one side, or to the regu
lar pentagon and dodecahedron on the other.

I)i media tutissimus ibis. The broad and beaten
middle highway is undoubtedly the safest for the multi
tude, who can thereby easily, although very indirectly,
ascend the hill of science to the table-lands on which
they are content to dwell, but for the expert climber who,
as an explorer, would reach, or, as a guide, would lead
others to the summit, the shorter, narrower, more nearly
direct and capable of being reduced to absolutely direct
path pointed out by nature, but hitherto wrholly un-
tracked, is, as it would seem to the author, far better
adapted, and if it had been sought would have been
readily found and, as it would also seem, unquestion
ably pursued.

Whether or not in these days of marvellous progress,
this path in its zigzag course or made straight by span
and trestle, viaduct, cut, and tunnel, will be opened as a
new highway, remains to be seen, but the author verily
believes that if not in these, it will be in later days.

QUESTIONS IN MATHEMATICS,

POSTSCRIPT.

Strike out, beginning with the 3d ff from tbe bottom of
page 70 to end of 7th line of page 107, and insert instead :

The greatest second and third powers contained in
any given number and the roots thereof respectively
may be found by considering the given number as the
area of a regular triangle with reference to the second
power and as the volume of a regular tetrahedron with
reference to the third, and analyzing those figures to
find the side of the triangle and the edge of the tetra
hedron, which are the second and third roots respec
tively.

Let it be required to find the side of a regular
triangle the area of which is given as 127449.

The figure consists of as many rows of regular
triangles of side 1 (including the initial triangle con
sidered as a row) as the number of the required side,
each row after the first exceeding the next preceding
row by 2.

There are three places of figures in the number of
the side, the first figure being 3, to which affix two
ciphers, making 300.

2 QUESTIONS IN MATHEMATICS.

From the given number
subtract the number of triangles contained in
the first 300 rows — 3002 = 90000

leaving remainder . . 37449

The difference between the numbers of tri
angles contained in the 300th and 310th rows
= (300 + 310) X 10 = 6100, and between
those contained in the 310th and 3^0th rows
= (310 + 320) X 10 == 6300, and so on, each
successive difference exceeding the next preced
ing by 200. Omitting the ciphers, the first five
differences and their sum are 61 -f- 63 -|- 65 -f-
67 + 69 = 60 X 5 + 52 = . . . . 326

which subtracted from the first three figures of
the remainder, as above, leaves second remain
der . . 4949

5, being the number of differences contained
in the first remainder is thus found to be the
second figure of the side.

The difference between the numbers of tri
angles contained in the 350th and 351st rows
= 350 4- 351 = 701. It will now be readily
seen that 7 is the third and last figure of the
side.

700 X 7 + (1 + 3 + 5 -f- 7 + 9 + 11 -f 13
= 72) = . . . . 4949

which subtract from second remainder, .. . ....

There being no third remainder, the given number
is a perfect second power, the root of which is 357.

Let the given number be 23109986.
There are four places of figures in the root. First
figure 4, to which affix three ciphers.

QUESTIONS IN MATHEMATICS.

Given number,

40002 =

40 -f 41 = 81.

80 X 8 + 8a =

which subtracted from first three figures of
remainder as above leaves second remainder,

8 is thus found to be the second figure of
the root.

480 + 481 ; = 961, which exceeds the first
three figures of the remainder.

The third figure of the root is thereby
found to be 0.

4800 + 4801 : = 9601.

9600 X 7 -|- 72 =

7 is thus found to be the fourth and last
figure of the root, making the whole root 4807.

Sum last found subtracted from remainder,
as before, leaves remainder of given number
over and above the greatest second power
therein contained,
which subtracted from given number,

gives such greatest second power, 4807a ^

23109986
16000000

7109986
704

69986

67249

2737
23109986

23107249

The volume in tetrahedra of edge 1 of a regular
tetrahedron of any given edge is equal to the product
of the number of octahedra of edge 1 contained in
such regular tetrahedron multiplied by 6 -4- the number
of the edge. Conversely, the number of octahedra of
edge 1 contained in a regular tetrahedron of any given
volume in tetrahedra of edge 1 is equal to the quo
tient of such volume divided by 0 — the quotient of
the edge of such regular tetrahedron divided by 6.

QUESTIONS IN MATHEMATICS.

Let it be required to rind the edge of a regular tetra
hedron the volume of which is given as 1367631.

There are three places of figures in the edge, and the
first figure of the edge is 1, to which affix two ciphers.

Find the quotient by 6 of the given vol-
1367631

ume.

The number of octahedra of edge 1 con
tained in a regular tetrahedron of edge 100 is

equal to 1002 x 1-— - — ° =
6 0

which, subtracted from the quotient as above,
leaves remainder . . . .

The difference between the numbers of oc
tahedra of edge 1 contained in two regular
tetrahedra of edges 100 and 110 respectively

( = 55 ) with three ciphers affixed

+ the number of such octahedra in a re
gular tetrahedron of edge 10 ( = 102 x

227938.5

166650

61288.5

55165

which, subtracted from the remainder as
above, leaves second remainder .

The like difference in respect to two
regular tetrahedra of edges 110 and 111 re
spectively is equal to the number of octa
hedra contained in course 111 of the latter

= 111- JJL0 (see last ^ on page 68)=

which subtracted from the second remainder
as above leaves third remainder . .
18.5 x 6 = 111, which is the required edge.

6105

18.5

QUESTIONS IN MATHEMATICS. 5

VERIFICATION.

The number of octahedra of edge I contained in the
regular tetrahedron which has been thus analyzed con
sists as follows :

Of the number in 100 .... 166650

+ the difference between 100 and 110 . . 55165
4- " " 110 " 111 . . 6105

Total number of octahedra . . . 227920

To find their volume in tetrahedra of edge
1, multiply the number by .... 4

Volume of the octahedra .... 911(580
To find the number of tetrahedra of edge
1 contained in such regular tetrahedron, mul
tiply the number of the octahedra by 2 and
to the product add the number of the edge
111 (see 6th 1 on page 69) = . 455951

1113 = . . . . 1367631

This, as will be readily seen, is equivalent to mul
tiplying the number of octahedra by 6 and adding to
the product the number of the edge.

Let the given number, the greatest third power and
remainder over, if any, contained in which are required,
and also the root of the power, be llllllllllll.

No. of places of figs, in root 4.

First fig. of root 4, to which afiix three ciphers, making
4000.

Quotient by 6 of given no. = . 18518518518.5

No. oct. in 4000 = 40002 x —t

^1 -- 10666666000

()

which subtracted from quo., as above,

leaves remainder . 7861862518.6

6 QUESTIONS ITS' MATHEMATICS.

To find the second figure of the root.
Diff. bet. nos. oct, in 4000 and 4100

41 v 40

- ( = 820 ) with 6 ciphers affixed

+ no. oct. in 100 (166650) = 820166650
+ diff. bet. 4100 and 4200

42 x 41

J*pL(=820+41 = 861)

j£

with 6 ciphers affixed + no.

oct. in 100, as before = 861166650

+ diff. bet. 4200 and 4300

= 861 + 42, etc., as before = 903166650

4- diff. bet. 4300 and 4400

= 903 + 43, etc., as before = 916166650

+ diff. bet. 4400 and 4500

= 946 + 44, etc., as before = 990166650

+ diff. bet. 4500 and 4600

= 990 + 45, etc., as before = 1035166650

+ diff. bet. 4600 and 4700

= 1035 + 46, etc., as before = 1081166650

+ diff. bet 4700 and 4800

= 1081 + 47, etc., as before = 1128166650 = 7765333200

= diff. bet. nos. oct. in 4000 and 4800,

which, subtracted from rem. as above,

leaves second rem. ..... 86519318.5

8, being the number of the differences
thus added together, is the second figure
of the root. Substitute same for the
first cipher affixed, making 4800.

To find the third figure of the root.
Diff. bet. nos. oct. in 4800 and 4810

= 481 x 48Q (= 115440) with 3 ciphers

affixed + no. oct. in 10 (165) = 115440165

QUESTIONS IN MATHEMATICS.

This difference being in excess of the
remainder, the third figure of the root is
thereby found to be 0. Eetain second
cipher affixed, making still 4800.

To find the fourth and last figure of
the root.

Diff. bet. nos. oct. in 4800 and 4801
4801 X 4800_ n

+ 11522400 + 4801
+ 11527201 + 4802
+ 1 1532003 + 4803
+ 1 1536806 + 4804
+ U541610 + 4805
+ 1 1546415 + 4806

11527201
11532003
11536806
11541610
11546415
11551221

= diff. bet. nos. oct. in 4800 and 4807,
which, subtracted from rem. as above,
leaves rem. ......

7, being the number of the differences
thus added together, is the fourth and
last figure of the root, and being substi
tuted for the last cipher affixed, makes
the whole root 4807.

Subtract quo. by 6 of whole root from
preceding remainder. - . .

= 80757656

5761662.5

801.166

5760861.333

Last remainder .

which restored to volume by multiplica
tion by . .

= remainder of given number over great
est third power therein contained . 34565168
Subtract same from given number 111111111111

Rem. = greatest third power required, 111076545943

8 QUESTIONS IN MATHEMATICS.

The foregoing example exhibits the process in its
full elaboration. But it may be very considerably
shortened as follows :

1st. In respect to finding no. oct. in 4000.

There is no octahedron, except fractional, in a regular

tetrahedron of edge 1. I2 X- - = 0.

6 0

No. oct. in 10 = 10* X — - — = 165.

6 6

No. oct. in 100 = no. oct. in 10 with 3 ciphers
affixed =.-.•;.: . 165000 .
+ no oct, in 10 X 10 = 1650 = 166650.

No. oct. in 1000 = no. oct. in 10 with 6 ciphers
affixed : . . . 165000000
+ no. oct. in 100 X 10 = 1666500 = 166666500

Thus it will be seen that the number of octahedra
contained in every regular tetrahedron, the first figure
of the edge of which is 1 and is followed by ciphers
throughout, may be found by inserting for each cipher
beyond the first two 6' s between 6 and 5 as found in 10,
and affixing for each such cipher one cipher.

Thus no. oct. in 10000 = 166666665000.

The number of octahedra contained in any regular
tetrahedron, the first figure of the edge of which is any
one of the other eight digits and is followed by ciphers
throughout, is equal to the number in a regular tetra
hedron of which the first figure is the edge with three
ciphers affixed thereto for each cipher in the edge
(but observe, including and not beyond the first cipher,
as before) + the -number in 10, 100, 1000, and so on (the
ciphers being the same in number as those in the edge)
X the first figure of the edge.

QUESTIONS IN MATHEMATICS. 9

Thus no. oct. in 20 = 2* X -?- ~ = 1 which with 3

6 6

ciphers affixed = .' , . ... 1000

+ no. oct. in 10 (165) X 2 = . . . 330

= 202 X — -• -— = 1330

6 6

No. oct in 300 = 32 X - - = 4 which with 6

6 6

ciphers affixed = 4000000

+ no, oct. in 100 (166650) X 3 = . . 499950

= 3002 X -— - 4499950

6 6

No. oct in 4000 = 42 X — — L = 10 which with 9

6 6

ciphers affixed = ..... 10000000000

-f no. oct. in 1000 (166666500) X 4 = . 666666000

= 40002 X -- — = 10666666000

6 6

as in the example.

2d. In respect to finding the second figure of the
root (8).

The first remainder in the example is 7851852518.5
and the first difference is 820166650.

The second figure of the root may be found by means
of the first four figures of the remainder and the first
three figures of the difference in connection with the
following series :

JVos. of terms. 123456789
Series. 0. 1. 3. 6. 10. 15. 21. 28. 36

Find the greatest term of the series, the product of
40 multiplied by which will, when increased by the

10 QUESTIONS IN MATHEMATICS.

sum of all the terms up to and including the term
and added to the product of the first three figures of
the difference multiplied by the number of the term,
yield a sum less than the first four figures of the
remainder.

4tf X 28 = . .'..'. 1120

+ sum of all the terms up to and including 28 = 84

+ 820 X 8 = . . . ' , . . 6560

~- ..-.-. . . 7764

which is less than 7851, the first four figures of the re
mainder. If 36, the 9th term of the series, had been
taken, the resulting sum would have been 8940, which
is greater than 7851. 28 is, therefore, the greatest term
required, and the number thereof, 8, is the second figure
of the root.

The rationale of this process is as follows :
By reference to the example it will be seen that the
first three figures of the second difference exceed those
of the first difference by 41 ; that those of the third
difference exceed those of the second by 42, and so on
up to the eighth difference in which the excess over the
seventh is 47. 40 is contained in the sum of these suc
cessive increments HO times, but 2 of these times (the
excess over 28) constitute part of that part of the sum
of the increments which results from the addition
together of the products of the units 1, 2, 3, etc., up to
7 multiplied l>y 7, 0, 5, etc., down to 1 respectively
= 84.

The series consists of the numbers of octahedra con
tained in the several courses of a regular tetrahedron
of edge 9, and the sum of the terms up to and including
any term is the number of octahedra contained in a
regular tetrahedron of which the number of the last
included term is the edge.

QUESTIONS IN MATHEMATICS.

3d. The next step of the process is to find the dif
ference between the numbers of octahedra contained in
two regular tetrahedra of edges: 4000 and 4800 (root
so far as found) respectively, which may be done as
follows :

Diff. bet. nos. oct. in 4000 and 4100 as

before found =

X second figure of root as found

to which add 40 X 28 (8th term of series)
= 1120 with 0 ciphers affixed =
-4- sum of series up to and including 8th
term = 84 with 0 ciphers affixed =

820166650
8

6501333200

1120000000

84000000

7765333200

= diff. bet. nos. oct. in 4000 and 4800 as found in the
example by the finding and addition together of the
eight successive differences between 4000 and 4100,
4100 and 42o(), and so on up to 4700 and 4800.

But the remainder and first difference, as before found,
may be such that it will at once be perceived what the
required figure of the root is without recourse to the
series, and in such case the required sum of all the dif
ferences may be found by a shorter process, which in
this example would be as follows :

Diff. bet. nos. oct. in 4000 and 4800 =

960 X 8 = 7680, which with 6 ciphers

affixed = . . 7680000000

-[-no. oct. in 800 = : no. oct. in 8 with 6

ciphers affixed = . 84000000

-t- no in 100 (166650) X 8 = 1333200 : 85333200

. 7765333200

as before and as in the example.

12 QUESTIONS IN MATHEMATICS.

4th. In respect to finding the fourth and last figure
of the root (7) and at the same time the difference be
tween nos. oct. in 4800 and 4807.

The remainder shown in the example is 86519318.5,
and the first difference is . . 11522400

and by comparison and consideration of
them in connection with the series, it will
be found that term 21 of the latter is the
required term, and that the number thereof
(7) is the fourth and last figure of the root.

Diff. as above x no. of required term 7

= ...-..'.... 80656800

-4- 4800 (as above) X 21, required term, = 100800
-4- sum of series up to and including term

21 : 56

. 80757656

= diff. bet. nos. oct. in 4800 and 4807 as
found in the example.

Or, thejast figure of the root being perceived by com
parison of the remainder and first difference found as
above to be 7, the whole difference may be found by
the shorter and direct process before shown, as follows :

Diff. bet. nos. oct. in 4800 and 4807 = 4807 * 480°

; 11536800

X 7

= . . 80757600

-f- no. oct. in 7 = . . .' . . . 56

: . . 80757656

Let the given number be 45499294.

There are three places of figures in the root, and the

first figure of the root is 3.

QUESTIONS IN MATHEMATICS. 13

45499294 = . . . . . . 7583218.666

— 'B* X 7 -7; with 6 ciphers affixed
6 b

+ 1CCG50 X 3 = 4499950.

... . . . . . . 3083265.666

31 X 30
— - — with 3 ciphers affixed

+ 10.") = . . • . . . 465165
X no. of term 10 of series, such no.
being the second figure of the root, _ 5

. . . . . 2325825
+ 30 X 10 with 3 ciphers affixed = 300000
+ 0+1 +3 + 6.'+ 10 with 3
ciphers affixed = . . 20000 = 2645825

437440.666
351 X 350

X no. of term 21 of series, such no.
being the third and last figure of
the root ..... _ 7

..... ; 429975
+ 350 X 21 = . . . . 7350

+ 0 + 1 + 3 + 6+10 + 15 + 21= 56 = 437381

59.666

357

- quo. by 6 of root as found, - 59.5

0.166

X . . . . 6

- rem. of given no. over greatest

third power therein contained . 1

which subtracted from given no. . 45499294

gives such greatest third power.

3573 = 45499293

14 QUESTIONS IN MATHEMATICS.

In the usual arithmetical process based upon an
analysis of the cube each step after the first is in the
first instance tentative ; but in the process based upon
an analysis of the regular tetrahedron there is no ten
tative step, but absolute certainty throughout.

Which of the two plane figures, the regular triangle
or the square, and of the two solid figures, the regular
tetrahedron or the cube, respectively, seems the better
adapted to finding by an analysis thereof the greatest
second or third power, respectively, the root thereof
and the remainder over, if any, contained in the given
area or volume of any figure, plane or solid, perfect or
imperfect ?

9
356

Smith, John C.

Questions in mathematics

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